Introduction to trigonometry

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レッスンノート

What Trigonometry Is

Trigonometry is the branch of mathematics that links the angles of a triangle to the lengths of its sides.
It began in the Hellenistic world in the 3rd century BC, growing out of geometry and the study of the stars.
Early mathematicians in India produced the first known tables of values for the sine ratio.
Trigonometry is used in surveying, navigation, geodesy and celestial mechanics.
In this topic we work only with right-angled triangles.

Labelling a Right-Angled Triangle

The hypotenuse is the side opposite the 90° angle. It is always the longest side.
Choose one of the two acute angles and call it A.
The opposite side is the side across from angle A — it does not touch A.
The adjacent side is the other side that touches angle A (it is not the hypotenuse).
The opposite and adjacent sides swap over if you switch to the other acute angle, but the hypotenuse never changes.
The opposite side is sometimes called the perpendicular and the adjacent side the base.

The Three Ratios: SOH CAH TOA

Sine: \sin A = \frac{\text{opposite}}{\text{hypotenuse}} — remember SOH.
Cosine: \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} — remember CAH.
Tangent: \tan A = \frac{\text{opposite}}{\text{adjacent}} — remember TOA.
Using letters a (opposite), b (adjacent), h (hypotenuse): \sin A = \frac{a}{h}, \cos A = \frac{b}{h}, \tan A = \frac{a}{b}.
The three ratios are linked: \tan A = \frac{\sin A}{\cos A}.
Each ratio depends only on the size of angle A, not on how big the triangle is — any two right-angled triangles with the same acute angle are similar.

Why the Ratios Stay the Same

Two right-angled triangles with the same acute angle are similar, so their sides are in the same proportion.
That is why \sin A, \cos A and \tan A give a single value for a given angle.
Because of this, the ratios define functions of the angle, called the trigonometric functions.
This means a calculator can return the same value for an angle no matter what triangle you draw.

Finding a Missing Side

Label the sides opposite, adjacent and hypotenuse relative to the angle you know.
Decide which ratio connects the side you know to the side you want — use SOH CAH TOA.
Write the equation, then rearrange it to make the unknown side the subject.
Substitute the angle and the known length, then use the sin, cos or tan button on your calculator.
Check your answer is sensible: the hypotenuse must be the longest side, and no side can be longer than the hypotenuse.

Finding a Missing Angle

Label the sides relative to the angle you are trying to find.
Pick the ratio that uses the two sides you know.
Form the ratio as a fraction, for example \tan A = \frac{\text{opposite}}{\text{adjacent}}.
Use the inverse function on your calculator: \sin^-1, \cos^-1 or \tan^-1.
Make sure your calculator is set to degrees before you start.
Check the answer is acute (less than 90°), since it is an angle in a right-angled triangle.

Angles of Elevation and Depression

An angle of elevation is measured upwards from the horizontal to an object above you.
An angle of depression is measured downwards from the horizontal to an object below you.
Both are measured from a horizontal line, never from a vertical one.
The angle of elevation from A to B equals the angle of depression from B to A — they are alternate angles.
These angles turn real situations (heights, distances, ladders, cliffs) into right-angled triangle problems.

Exact Values for 30°, 45° and 60°

\sin 30° = \frac{1}{2}, \cos 30° = \frac{\√{3}}{2}, \tan 30° = \frac{1}{\√{3}}.
\sin 45° = \frac{\√{2}}{2}, \cos 45° = \frac{\√{2}}{2}, \tan 45° = 1.
\sin 60° = \frac{\√{3}}{2}, \cos 60° = \frac{1}{2}, \tan 60° = \√{3}.
Notice \sin 30° = \cos 60° and \sin 60° = \cos 30° — the sine of an angle equals the cosine of its complement.
These exact values come from the special triangles: the half-equilateral triangle (30°–60°–90°) and the isosceles right-angled triangle (45°–45°–90°).

Reciprocal Ratios (Extension)

The cosecant is the reciprocal of sine: \csc A = \frac{1}{\sin A} = \frac{\text{hypotenuse}}{\text{opposite}}.
The secant is the reciprocal of cosine: \sec A = \frac{1}{\cos A} = \frac{\text{hypotenuse}}{\text{adjacent}}.
The cotangent is the reciprocal of tangent: \cot A = \frac{1}{\tan A} = \frac{\text{adjacent}}{\text{opposite}}.
These are useful for rewriting trigonometric expressions into a more convenient form.

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練習問題

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1.In a right-angled triangle, what is the name of the longest side?
Easy
- Ahypotenuse
- Badjacent
- Copposite
- Dperpendicular
2.Which ratio is defined as opposite ÷ hypotenuse?
Easy
- Asin A
- Bcos A
- Ctan A
- Dsec A
3.The tangent of an angle in a right-angled triangle is equal to the opposite side divided by the adjacent side.
Easy
True or false?
4.Which of the following are trigonometric ratios? (select all that apply)
Medium
- Asine
- Bcosine
- Ctangent
- Dsecant
- Ecosecant
5.Match each trigonometric ratio with its correct definition.
Medium
- sin A
- cos A
- tan A
- opposite ÷ hypotenuse
- adjacent ÷ hypotenuse
- opposite ÷ adjacent
6.Arrange the steps to find a missing side in a right-angled triangle using trigonometry.
Easy
- Label the sides
- Choose the correct ratio
- Set up the equation
- Solve for the unknown side
7.Which trigonometric ratio should be used to find the length of the opposite side?
Medium
- Atan
- Bsin
- Ccos
- Dsec
8.Which of these is the correct definition of the cosecant ratio?
Easy
- Acsc A = hypotenuse ÷ opposite
- Bcsc A = opposite ÷ hypotenuse
- Ccsc A = hypotenuse ÷ adjacent
- Dcsc A = adjacent ÷ hypotenuse

もっと問題

9.Match each trigonometric function to its reciprocal.
Easy
- sin A
- cos A
- tan A
- cosec A
- sec A
- cot A
10.Which of the following are true for a right-angled triangle with an acute angle A? (select all that apply)
Medium
- Asin A = opposite / hypotenuse
- Bcos A = adjacent / hypotenuse
- Ctan A = adjacent / opposite
- Dtan A = sin A / cos A
- Esin A = adjacent / hypotenuse
11.Arrange the steps in the correct order to find the length of the opposite side when you know the hypotenuse and an acute angle in a right-angled triangle.
Medium
- Identify the opposite side and hypotenuse relative to the given angle.
- Choose the sine ratio because it links opposite and hypotenuse.
- Substitute the known values into sin(angle) = opposite / hypotenuse.
- Solve for the opposite side by multiplying both sides by the hypotenuse.
12.In a right-angled triangle, the side next to a given angle that is not the hypotenuse is called the:
Medium
- Aopposite
- Badjacent
- Chypotenuse
- Dperpendicular
13.The hypotenuse is always the longest side of a right-angled triangle.
Easy
True or false?
14.Which trigonometric ratio should be used to find the length of the hypotenuse when you know the opposite side and an acute angle?
Medium
- Asine
- Bcosine
- Ctangent
- Dcosecant
15.Which of the following statements about the reciprocals of trigonometric ratios are correct? (select all that apply)
Medium
- Acosec A = hypotenuse / opposite
- Bsec A = hypotenuse / adjacent
- Ccot A = adjacent / opposite
- Dcosec A = opposite / hypotenuse
- Esec A = adjacent / hypotenuse
16.Match each angle with its exact trigonometric value.
Medium
- sin 30°
- cos 45°
- tan 60°
- 1/2
- √2/2
- √3
17.Arrange the steps in the correct order to find a missing angle in a right-angled triangle when you know the opposite and adjacent sides.
Medium
- Label the sides opposite and adjacent to the unknown angle.
- Choose the tangent ratio because it links opposite and adjacent.
- Substitute the known side lengths into tan(angle) = opposite / adjacent.
- Use the inverse tangent function to find the angle.
18.In a right-angled triangle, angle A is acute. Which expression gives the value of tan A?
Easy
- A\frac{\text{opposite}}{\text{adjacent}}
- B\frac{\text{adjacent}}{\text{hypotenuse}}
- C\frac{\text{opposite}}{\text{hypotenuse}}
- D\frac{\text{hypotenuse}}{\text{opposite}}
19.A right-angled triangle has an angle of 35°. The side adjacent to this angle is 8 cm. Which calculation gives the length of the opposite side, correct to 2 decimal places?
Medium
- A8 × tan 35° = 5.60 cm
- B8 ÷ tan 35° = 11.43 cm
- C8 × sin 35° = 4.59 cm
- D8 × cos 35° = 6.55 cm
20.A right-angled triangle has an angle of 50°. The side opposite this angle is 7 cm. Which calculation gives the length of the hypotenuse, correct to 2 decimal places?
Medium
- A7 ÷ sin 50° = 9.14 cm
- B7 × sin 50° = 5.36 cm
- C7 ÷ cos 50° = 10.89 cm
- D7 × tan 50° = 8.34 cm
21.A right-angled triangle has an angle of 28°. The hypotenuse is 15 cm. Which calculation gives the length of the side adjacent to the 28° angle, correct to 2 decimal places?
Medium
- A15 × cos 28° = 13.24 cm
- B15 × sin 28° = 7.04 cm
- C15 ÷ cos 28° = 16.99 cm
- D15 × tan 28° = 7.98 cm
22.In a right-angled triangle, the side opposite angle A is 9 cm and the hypotenuse is 15 cm. Which calculation gives the size of angle A, correct to 1 decimal place?
Medium
- Asin^-1(9/15) = 36.9°
- Bcos^-1(9/15) = 53.1°
- Ctan^-1(9/15) = 31.0°
- Dsin^-1(15/9) = 56.4°
23.In a right-angled triangle, the side opposite angle A is 5 cm and the side adjacent to angle A is 12 cm. Which calculation gives the size of angle A, correct to 1 decimal place?
Medium
- Atan^-1(5/12) = 22.6°
- Btan^-1(12/5) = 67.4°
- Csin^-1(5/12) = 24.6°
- Dcos^-1(5/12) = 65.4°
24.Which of the following calculations are correct for a right-angled triangle with an acute angle of 40° and a hypotenuse of 10 cm? (select all that apply)
Medium
- Aopposite = 10 × sin 40° ≈ 6.43 cm
- Badjacent = 10 × cos 40° ≈ 7.66 cm
- Copposite = 10 ÷ sin 40° ≈ 15.56 cm
- Dadjacent = 10 ÷ cos 40° ≈ 13.05 cm
- Eopposite = 10 × tan 40° ≈ 8.39 cm
25.For a right-angled triangle with an acute angle of 60° and an opposite side of 6 cm, which of the following statements are correct? (select all that apply)
Medium
- AThe hypotenuse is 6 ÷ sin 60° ≈ 6.93 cm
- BThe adjacent side is 6 ÷ tan 60° ≈ 3.46 cm
- CThe hypotenuse is 6 × sin 60° ≈ 5.20 cm
- DThe adjacent side is 6 × tan 60° ≈ 10.39 cm
- EThe hypotenuse is 6 ÷ cos 60° = 12 cm
26.In a right-angled triangle, the side opposite the 90° angle is called the hypotenuse.
Easy
True or false?
27.For any acute angle A in a right-angled triangle, sin A = cos A.
Easy
True or false?
28.Match each trigonometric ratio with the correct formula for an acute angle A in a right-angled triangle.
Medium
- sin A
- cos A
- tan A
- \frac{\text{opposite}}{\text{hypotenuse}}
- \frac{\text{adjacent}}{\text{hypotenuse}}
- \frac{\text{opposite}}{\text{adjacent}}
29.Arrange the steps in the correct order to find the length of the adjacent side in a right-angled triangle when you know the hypotenuse and an acute angle.
Medium
- Identify the hypotenuse and the adjacent side relative to the given angle.
- Choose the cosine ratio: cos A = adjacent / hypotenuse.
- Substitute the known values into the equation.
- Rearrange to make the adjacent side the subject.
- Calculate the value using a calculator and round as required.
30.A ladder 8 m long leans against a vertical wall. The foot of the ladder is 3 m from the base of the wall. Which calculation gives the angle the ladder makes with the ground, correct to 1 decimal place?
Medium
- A\cos^-1(3/8) = 68.0°
- B\sin^-1(3/8) = 22.0°
- C\tan^-1(3/8) = 20.6°
- D\cos^-1(8/3) is undefined
31.From a point 40 m from the foot of a vertical tower, the angle of elevation of the top of the tower is 32°. Which calculation gives the height of the tower, correct to 2 decimal places?
Medium
- A40 × tan 32° = 24.99 m
- B40 × sin 32° = 21.20 m
- C40 ÷ tan 32° = 64.01 m
- D40 × cos 32° = 33.92 m
32.A ramp rises at a constant angle of 12° to the horizontal and its sloping surface is 5 m long. Which calculation gives the vertical height gained at the top of the ramp, correct to 2 decimal places?
Medium
- A5 × sin 12° = 1.04 m
- B5 × cos 12° = 4.89 m
- C5 × tan 12° = 1.06 m
- D5 ÷ sin 12° = 24.05 m
33.A kite is held on a string 25 m long. The string makes an angle of 38° with the horizontal. Which calculation gives the height of the kite above the hand holding the string, correct to 2 decimal places?
Hard
- A25 × sin 38° = 15.39 m
- B25 × cos 38° = 19.70 m
- C25 × tan 38° = 19.53 m
- D25 ÷ tan 38° = 32.00 m
34.A ship sails 6 km due north, then 8 km due east. Which calculation gives the angle between the ship's final position and its starting point, measured from the north direction, correct to 1 decimal place?
Hard
- A\tan^-1(8/6) = 53.1°
- B\tan^-1(6/8) = 36.9°
- C\sin^-1(8/6) is undefined
- D\cos^-1(6/8) = 41.4°
35.A vertical flagpole stands on horizontal ground. From a point 15 m from its base, the angle of elevation of the top of the pole is 48°. Which of the following statements are correct? (select all that apply)
Hard
- AThe height of the pole is 15 × tan 48° = 16.66 m to 2 decimal places.
- BThe height of the pole is 15 × sin 48° = 11.15 m to 2 decimal places.
- CThe angle of depression from the top of the pole to the point on the ground is 48°.
- DThe height of the pole is 15 ÷ tan 48° = 13.51 m to 2 decimal places.
- EThe distance from the point to the top of the pole is \√{15² + 16.66²} m.
36.A surveyor measures a right-angled triangle with a 35° angle. The side opposite the 35° angle is 12 m. Which of the following quantities are correct to 2 decimal places? (select all that apply)
Hard
- AHypotenuse = 12 ÷ sin 35° = 20.92 m
- BAdjacent side = 12 ÷ tan 35° = 17.14 m
- CHypotenuse = 12 × sin 35° = 6.88 m
- DAdjacent side = 12 × tan 35° = 8.40 m
- EHypotenuse = 12 × cos 35° = 9.83 m
37.The angle of elevation from a point on the ground to the top of a tower is equal to the angle of depression from the top of the tower to that same point.
Medium
True or false?
38.In a right-angled triangle, if the opposite side is doubled while the angle stays the same, the hypotenuse also doubles.
Medium
True or false?
39.Match each real-life situation with the trigonometric ratio that finds the unknown directly.
Medium
- Height of a tower from its shadow and the Sun's angle of elevation
- Horizontal distance from a string length and the angle of elevation
- Angle of a ramp from its rise and run
- tan
- sin
- cos
40.Arrange the steps in the correct order to find the height of a vertical tower when the angle of elevation from a point 30 m away is 50°.
Medium
- Identify the height as the opposite side and 30 m as the adjacent side.
- Write tan 50° = height ÷ 30.
- Rearrange to height = 30 × tan 50°.
- Evaluate to get height = 35.75 m to 2 decimal places.
41.Complete the definition of cosine.
Easy
cos A = ____ ÷ ____
42.What mnemonic is commonly used to remember the trigonometric ratios sine, cosine and tangent?
Easy
43.A right-angled triangle has an angle of 30°. The side opposite this angle is 5 cm. Calculate the length of the hypotenuse, giving your answer to 2 decimal places.
Medium
44.Calculate the length of the opposite side, giving your answer to 2 decimal places.
Medium
45.A right-angled triangle has an angle of 40°. The hypotenuse is 12 cm. Calculate the length of the side opposite the 40° angle, giving your answer to 2 decimal places.
Medium
46.Complete the definition of tangent.
Easy
tan A = ____ / ____
47.Complete the definition of the sine ratio.
Easy
sin A = ____ ÷ ____
48.A right-angled triangle has an angle of 25°. The hypotenuse is 14 cm. Complete the calculation for the side opposite the 25° angle, correct to 2 decimal places.
Medium
opposite = 14 × sin 25° = ____ cm
49.A right-angled triangle has an angle of 30°. The hypotenuse is 10 cm. Calculate the length of the side opposite the 30° angle, giving your answer in cm correct to 2 decimal places.
Medium
50.A right-angled triangle has an angle of 45°. The hypotenuse is 12 cm. Calculate the length of the side adjacent to the 45° angle, giving your answer in cm correct to 2 decimal places.
Medium
51.A right-angled triangle has an angle of 55°. The side adjacent to this angle is 9 cm. Calculate the length of the opposite side, giving your answer in cm correct to 2 decimal places.
Medium
52.Name the side of a right-angled triangle that is opposite the right angle.
Easy
53.Explain how you would find the size of angle A in a right-angled triangle given the lengths of the opposite and adjacent sides.
Hard
54.Explain how to find the length of the adjacent side in a right-angled triangle when you know the hypotenuse and an acute angle.
Medium
55.Explain how you would find the length of the opposite side in a right-angled triangle when you know the hypotenuse and an acute angle.
Medium
56.Explain how you would find the size of an acute angle in a right-angled triangle when you know the lengths of the opposite and adjacent sides.
Medium
57.A vertical pole 7 m tall casts a shadow of length 11 m on horizontal ground. Explain how you would find the angle of elevation of the Sun, and state the value to 1 decimal place.
Hard
58.A right-angled triangle has an angle of 30°. The side opposite this angle is 5 cm. Calculate the length of the hypotenuse, giving your answer to 2 decimal places.
Medium
- A12 cm
- B10 cm
- C11 cm
- D9 cm
59.Calculate the length of the opposite side, giving your answer to 2 decimal places.
Medium
- A7 cm
- B8 cm
- C6 cm
- D9 cm
60.A right-angled triangle has an angle of 40°. The hypotenuse is 12 cm. Calculate the length of the side opposite the 40° angle, giving your answer to 2 decimal places.
Medium
- A9.71 cm
- B8.71 cm
- C6.71 cm
- D7.71 cm
61.A right-angled triangle has an angle of 30°. The hypotenuse is 10 cm. Calculate the length of the side opposite the 30° angle, giving your answer in cm correct to 2 decimal places.
Medium
- A6 cm
- B7 cm
- C5 cm
- D4 cm
62.A right-angled triangle has an angle of 45°. The hypotenuse is 12 cm. Calculate the length of the side adjacent to the 45° angle, giving your answer in cm correct to 2 decimal places.
Medium
- A7.49 cm
- B10.49 cm
- C8.49 cm
- D9.49 cm
63.A right-angled triangle has an angle of 55°. The side adjacent to this angle is 9 cm. Calculate the length of the opposite side, giving your answer in cm correct to 2 decimal places.
Medium
- A14.85 cm
- B12.85 cm
- C13.85 cm
- D11.85 cm

復習ノート

## Introduction to Trigonometry Trigonometry studies the relationships between angles and side lengths in triangles. For right-angled triangles, the trigonometric functions give ratios of side lengths that depend only on one acute angle. ### Labelling a right-angled triangle - **Hypotenuse** — the side opposite the 90° angle; the longest side. - **Opposite** — the side opposite the angle you are working with. - **Adjacent** — the side next to the angle that is not the hypotenuse. Label relative to the angle you are given or asked about. ### The three ratios (SOH CAH TOA) Let A be an acute angle, with opposite = a, adjacent = b, hypotenuse = h. - \(\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{h}\) - \(\cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{h}\) - \(\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} = \frac{\sin A}{\cos A}\) **SOH CAH TOA** reminds you which ratio uses which pair of sides. ### Finding a missing side 1. Label the sides relative to the given angle. 2. Choose the ratio linking the known side and the side you want. 3. Substitute and solve. Use sin^{-1}, cos^{-1} or tan^{-1} only when finding an angle. ### Finding a missing angle Use the inverse functions on your calculator: - A = sin^{-1}(a/h) - A = cos^{-1}(b/h) - A = tan^{-1}(a/b) ### Angles of elevation and depression - **Angle of elevation** — measured upwards from the horizontal. - **Angle of depression** — measured downwards from the horizontal. ### Exact values (extension) | Angle | sin | cos | tan | |---|---|---|---| | 30° | 1/2 | √3/2 | 1/√3 | | 45° | √2/2 | √2/2 | 1 | | 60° | √3/2 | 1/2 | √3 | ### Reciprocals (for reference) - csc A = 1 / sin A = h/a - sec A = 1 / cos A = h/b - cot A = 1 / tan A = b/a ### Exam tips - Always check your calculator is in **degree mode**. - Sketch and label the triangle before choosing a ratio. - Round only at the final step.

Trigonometry: A branch of mathematics concerned with the relationships between angles and side lengths of triangles.
Hypotenuse: The side opposite the 90° angle in a right triangle; the longest side.
Opposite: The side opposite the angle you are working with.
Adjacent: The side next to the angle that is not the hypotenuse.
Sine (sin): The ratio of the side opposite the angle to the hypotenuse: sin A = opposite / hypotenuse.
Cosine (cos): The ratio of the adjacent side to the hypotenuse: cos A = adjacent / hypotenuse.
Tangent (tan): The ratio of the opposite side to the adjacent side: tan A = opposite / adjacent = sin A / cos A.
SOH CAH TOA: A mnemonic for the three ratios: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
Angle of elevation: The angle measured upwards from the horizontal.
Angle of depression: The angle measured downwards from the horizontal.
Cosecant (csc): The reciprocal of sine: csc A = 1 / sin A = hypotenuse / opposite.
Secant (sec): The reciprocal of cosine: sec A = 1 / cos A = hypotenuse / adjacent.
Cotangent (cot): The reciprocal of tangent: cot A = 1 / tan A = adjacent / opposite.

フラッシュカード

In a right-angled triangle, which side is the hypotenuse?

The hypotenuse is the side opposite the 90° angle. It is the longest side and is one of the two sides next to angle A.

How do you identify the opposite side relative to an acute angle A?

The opposite side is the side directly across from angle A. It does not touch angle A.

How do you identify the adjacent side relative to an acute angle A?

The adjacent side is the side that joins angle A to the right angle. It is the other side next to angle A, not the hypotenuse.

Write the sine ratio for an acute angle A in a right-angled triangle.

sin A = opposite / hypotenuse. In symbols: \sin A = \frac{\text{opposite}}{\text{hypotenuse}}.

Write the cosine ratio for an acute angle A in a right-angled triangle.

cos A = adjacent / hypotenuse. In symbols: \cos A = \frac{\text{adjacent}}{\text{hypotenuse}}.

Write the tangent ratio for an acute angle A in a right-angled triangle.

tan A = opposite / adjacent. In symbols: \tan A = \frac{\text{opposite}}{\text{adjacent}}.

What does the mnemonic SOH CAH TOA stand for?

SOH: sin = opposite / hypotenuse. CAH: cos = adjacent / hypotenuse. TOA: tan = opposite / adjacent.

Why do trigonometric ratios depend only on the size of the acute angle, not on the size of the triangle?

Any two right-angled triangles with the same acute angle are similar. Similar triangles have the same side ratios, so the ratios depend only on the angle.

How do you find a missing side in a right-angled triangle when you know one acute angle and one side?

Choose the ratio that links the known side and the missing side to the angle. Set up the equation, then solve it using your calculator.

How do you find a missing acute angle in a right-angled triangle when you know two sides?

Use the inverse trigonometric function: sin^-1, cos^-1 or tan^-1. For example, if sin A = 0.5, then A = sin^-1(0.5) = 30°.

What is an angle of elevation?

The angle of elevation is the angle measured upwards from the horizontal to an object above the observer.

What is an angle of depression?

The angle of depression is the angle measured downwards from the horizontal to an object below the observer.

State the exact value of sin 30°.

sin 30° = 1/2.

State the exact value of cos 30°.

cos 30° = √3 / 2.

State the exact value of tan 30°.

tan 30° = 1 / √3, which can also be written as √3 / 3.

State the exact value of sin 45°.

sin 45° = √2 / 2, which equals 1 / √2.

State the exact value of cos 45°.

cos 45° = √2 / 2, which equals 1 / √2.

State the exact value of tan 45°.

tan 45° = 1.

State the exact value of sin 60°.

sin 60° = √3 / 2.

State the exact value of cos 60°.

cos 60° = 1/2.

State the exact value of tan 60°.

tan 60° = √3.

What is trigonometry?

Trigonometry is the branch of mathematics that studies relationships between angles and side lengths of triangles. In right-angled triangles, trigonometric functions relate angles to ratios of side lengths.

Which fields use trigonometry?

Trigonometry is used in geodesy, surveying, celestial mechanics and navigation.

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