Trigonometric Graphs And Equations

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Trigonometric Graphs

Trig graphs are the graphs of y = sin x, y = cos x, and y = tan x for angles from 0° to 360°.
They have periodic shapes and symmetries that are essential to know.
The period of sin x and cos x is 360°; the period of tan x is 180°.

Graph of y = sin x

Oscillates between 1 and -1; passes through (0,0).
Key points: (0°,0), (90°,1), (180°,0), (270°,-1), (360°,0).
The graph is symmetric about the origin (odd function).

Graph of y = sin x

Graph of y = cos x

Oscillates between 1 and -1; y-intercept at (0,1).
Key points: (0°,1), (90°,0), (180°,-1), (270°,0), (360°,1).
y = cos x is a translation of y = sin x by 90° to the left.

Graph of y = cos x

Graph of y = tan x

Consists of branches separated by asymptotes at x = 90°, 270°, etc.
Passes through (0,0); each branch goes from -∞ to +∞.
Period is 180°; key points: (0°,0), (45°,1), (135°,-1), (180°,0), (225°,1), (315°,-1), (360°,0).

Graph of y = tan x

Solving Trig Equations

Use the inverse trig function on a calculator to find the first solution.
Use the graph of the trig function to find additional solutions within the given interval (usually 0° ≤ x ≤ 360°).
For sin x = k: if x is an acute solution, then 180° - x is the obtuse solution.
For cos x = k: if x is a solution, then 360° - x is the other solution.
For tan x = k: if x is a solution, then x + 180° is another solution.
Always check solutions by substituting back into the original equation.

Solving sin x = 0.5 using symmetry

Rearranging Trig Equations

Equations may need rearranging first, e.g., 2 sin x - 1 = 0 becomes sin x = 1/2.
Then solve using the standard method for the basic trig equation.

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1.What is the period of the graph of y = sin x?
Easy
- A90°
- B180°
- C360°
- D720°
2.What is the y-coordinate of the point where the graph of y = cos x crosses the y-axis?
Easy
- A0
- B1
- C-1
- D0.5
3.The graph of y = tan x has asymptotes at which x-values in the range 0° ≤ x ≤ 360°?
Easy
- A0° and 180°
- B90° and 270°
- C180° and 360°
- D0° and 360°
4.If sin x° = 0.36, find the acute angle x.
Easy
- A21.1°
- B36.0°
- C0.36°
- D69.0°
5.If sin x° = 0.36, find the obtuse angle x in the range 0° ≤ x ≤ 360°.
Medium
- A158.9°
- B201.1°
- C338.9°
- D21.1°
6.Solve tan x = 2 for 0° ≤ x ≤ 360°. Give the two solutions.
Medium
- A63.4° and 243.4°
- B63.4° and 116.6°
- C26.6° and 206.6°
- D63.4° and 296.6°
7.Solve 3 tan x = -4 for 0° ≤ x ≤ 360°. Give the two solutions.
Medium
- A126.9° and 306.9°
- B53.1° and 233.1°
- C126.9° and 233.1°
- D53.1° and 306.9°
8.x° is an obtuse angle and sin x° = 0.43. Find the value of x.
Medium
- A25.5°
- B154.5°
- C205.5°
- D334.5°

Thêm câu hỏi

9.Solve 3 cos x = 1 for 0° ≤ x ≤ 360°. Give the two solutions correct to 1 decimal place.
Medium
- A70.5° and 289.5°
- B70.5° and 109.5°
- C19.5° and 340.5°
- D19.5° and 160.5°
10.If cos α° = k, what is the value of cos (180° - α°)?
Medium
- A1 - k
- Bk
- C-k
- D-1 - k
11.If cos α° = k, what is the value of cos (360° + α°)?
Medium
- Ak - 1
- Bk + 1
- C-k
- Dk
12.Solve 4 sin x - 1 = 2 for 0° ≤ x ≤ 360°. Give the two solutions.
Medium
- A48.6° and 131.4°
- B48.6° and 311.4°
- C41.8° and 138.2°
- D41.8° and 318.2°
13.Solve 4 cos x + 2 = 3 for 0° ≤ x ≤ 360°. Give the two solutions.
Medium
- A75.5° and 284.5°
- B75.5° and 104.5°
- C60° and 300°
- D60° and 120°
14.Solve 5 tan x = -7 for 0° ≤ x ≤ 360°. Give the two solutions.
Medium
- A125.5° and 305.5°
- B54.5° and 234.5°
- C125.5° and 234.5°
- D54.5° and 305.5°
15.y is an obtuse angle. Which statement is true?
Easy
- Asin y > 0 and cos y > 0
- Bsin y > 0 and cos y < 0
- Csin y < 0 and cos y > 0
- Dsin y < 0 and cos y < 0
16.The graph of y = sin x is translated 90° to the left to obtain which graph?
Medium
- Ay = sin x + 90
- By = sin (x + 90)
- Cy = cos x
- Dy = tan x
17.What is the period of the graph of y = cos x?
Easy
- A90°
- B180°
- C360°
- D720°
18.What is the maximum value of y = sin x?
Easy
- A0
- B1
- C-1
- D2
19.At which x-value does the graph of y = tan x have an asymptote in the range 0° ≤ x ≤ 360°?
Easy
- A0°
- B90°
- C180°
- D360°
20.Solve sin x = 0.5 for 0° ≤ x ≤ 360°.
Easy
- A30° and 150°
- B30° and 210°
- C60° and 120°
- D60° and 300°
21.Solve cos x = -0.5 for 0° ≤ x ≤ 360°.
Easy
- A120° and 240°
- B60° and 300°
- C120° and 300°
- D60° and 120°
22.Solve tan x = 1 for 0° ≤ x ≤ 360°.
Easy
- A45° and 225°
- B45° and 135°
- C45° and 315°
- D135° and 315°
23.Solve 2 sin x = 1 for 0° ≤ x ≤ 360°.
Easy
- A30° and 150°
- B30° and 210°
- C60° and 120°
- D60° and 300°
24.Solve 3 cos x = 2 for 0° ≤ x ≤ 360°. Give answers correct to 1 decimal place.
Medium
- A48.2° and 311.8°
- B48.2° and 131.8°
- C41.8° and 318.2°
- D41.8° and 138.2°
25.Solve 4 tan x = -5 for 0° ≤ x ≤ 360°. Give answers correct to 1 decimal place.
Medium
- A128.7° and 308.7°
- B51.3° and 231.3°
- C128.7° and 231.3°
- D51.3° and 308.7°
26.The graph of y = sin x is translated 90° to the left. Which function does it become?
Easy
- Ay = cos x
- By = sin (x + 90°)
- Cy = sin (x - 90°)
- Dy = -cos x
27.Solve 5 sin x - 2 = 1 for 0° ≤ x ≤ 360°. Give answers correct to 1 decimal place.
Medium
- A36.9° and 143.1°
- B36.9° and 216.9°
- C53.1° and 126.9°
- D53.1° and 306.9°
28.Solve 2 cos x + 1 = 0 for 0° ≤ x ≤ 360°.
Medium
- A120° and 240°
- B60° and 300°
- C120° and 300°
- D60° and 240°
29.What is the period of the graph of y = tan x?
Easy
- A90°
- B180°
- C360°
- D720°
30.What is the y-intercept of the graph of y = cos x?
Easy
- A(0, 0)
- B(0, 1)
- C(0, -1)
- D(0, 0.5)
31.What is the gradient of the graph of y = sin x at x = 90°?
Medium
- A0
- B1
- C-1
- Dundefined
32.Solve 2 cos x = 1 for 0° ≤ x ≤ 360°. Give the two solutions.
Medium
- A60° and 300°
- B30° and 150°
- C60° and 120°
- D30° and 330°
33.The graph of y = sin x is reflected in the x-axis and then translated 90° to the left. Which function does it become?
Medium
- Ay = -cos x
- By = cos x
- Cy = -sin x
- Dy = sin x
34.The graph of y = cos x has a y-intercept of 1.
Easy
True or false?
35.Which of the following statements about the graph of y = sin x are true? (Select all that apply)
Medium
- AIt passes through (0, 0).
- BIts maximum value is 1.
- CIt has asymptotes at x = 90° and x = 270°.
- DIt is periodic with period 360°.
36.Match each trigonometric function to its y-intercept.
Medium
- y = sin x
- y = cos x
- y = tan x
- 0
- 1
- 0
37.Arrange the following x-values in the order they appear on the graph of y = sin x from x = 0° to x = 360°: 90°, 180°, 270°, 360°.
Easy
- 90°
- 180°
- 270°
- 360°
38.The equation tan x = 1 has exactly two solutions in the interval 0° ≤ x ≤ 360°.
Medium
True or false?
39.What is the y-coordinate of the point where the graph of y = tan x crosses the y-axis?
Easy
- A0
- B1
- C-1
- Dundefined
40.Solve 2 cos x = -√3 for 0° ≤ x ≤ 360°. Which are the two solutions?
Medium
- A150°, 210°
- B30°, 330°
- C120°, 240°
- D60°, 300°
41.The period of the graph of y = tan x is 360°.
Easy
True or false?
42.Arrange the following x-values in the order they appear on the graph of y = cos x from x = 0° to x = 360°: 90°, 180°, 270°, 360°.
Easy
- 90°
- 180°
- 270°
- 360°
43.Solve 2 cos x + √3 = 0 for 0° ≤ x ≤ 360°. Which are the two solutions?
Medium
- A150°, 210°
- B120°, 240°
- C30°, 330°
- D150°, 330°
44.Solve sin x = 0.5 for 0° ≤ x ≤ 360°. Give the smaller solution.
Easy
45.Solve cos x = -0.5 for 0° ≤ x ≤ 360°. Give the larger solution.
Easy
46.Solve tan x = 1 for 0° ≤ x ≤ 360°. Give the smaller solution.
Easy
47.Solve 2 sin x = 1 for 0° ≤ x ≤ 360°. Give the smaller solution.
Medium
48.Solve 2 cos x + 1 = 0 for 0° ≤ x ≤ 360°. Give the smaller solution.
Medium
49.Solve the equation 2 sin x = 1 for 0° ≤ x ≤ 360°. Give the two solutions, separated by a comma.
Medium
The solutions are ____ and ____.
50.Solve 3 tan x = -4 for 0° ≤ x ≤ 360°. Give the smaller solution correct to 1 decimal place.
Medium
51.Solve sin x = 0.5 for 0° ≤ x ≤ 360°. Give the obtuse solution.
Easy
52.Complete the sentence.
Easy
The graph of y = cos x has a y-intercept of ____.
53.Solve 3 tan x = -5 for 0° ≤ x ≤ 360°. Give the smaller solution correct to 1 decimal place.
Medium
54.State the number of solutions to the equation sin x = 0.5 in the interval 0° ≤ x ≤ 360°.
Easy
The equation sin x = 0.5 has ____ solutions in the interval 0° ≤ x ≤ 360°.
55.Write down the two solutions to the equation cos x = -0.5 for 0° ≤ x ≤ 360°. Give your answers in degrees.
Medium
56.Describe how to use the graph of y = sin x to find all solutions to sin x = 0.6 in the interval 0° ≤ x ≤ 360°. Give the solutions correct to 1 decimal place.
Hard
57.Solve sin x = 0.5 for 0° ≤ x ≤ 360°. Give the smaller solution.
Easy
- A27 °
- B36 °
- C30 °
- D33 °
58.Solve cos x = -0.5 for 0° ≤ x ≤ 360°. Give the larger solution.
Easy
- A264 °
- B240 °
- C288 °
- D216 °
59.Solve tan x = 1 for 0° ≤ x ≤ 360°. Give the smaller solution.
Easy
- A45 °
- B55 °
- C40 °
- D50 °
60.Solve 2 sin x = 1 for 0° ≤ x ≤ 360°. Give the smaller solution.
Medium
- A30
- B27
- C33
- D36
61.Solve 2 cos x + 1 = 0 for 0° ≤ x ≤ 360°. Give the smaller solution.
Medium
- A120
- B132
- C108
- D144
62.Solve 3 tan x = -4 for 0° ≤ x ≤ 360°. Give the smaller solution correct to 1 decimal place.
Medium
- A152.3 °
- B114.2 °
- C139.6 °
- D126.9 °
63.Solve sin x = 0.5 for 0° ≤ x ≤ 360°. Give the obtuse solution.
Easy
- A180 °
- B135 °
- C165 °
- D150 °
64.Solve 3 tan x = -5 for 0° ≤ x ≤ 360°. Give the smaller solution correct to 1 decimal place.
Medium
- A145.1 °
- B133 °
- C108.8 °
- D121.0 °

Ghi chú ôn tập

## Trigonometric Graphs & Equations ### Key Graphs - **y = sin x**: Wave oscillating between -1 and 1, period 360°. Key points: (0,0), (90°,1), (180°,0), (270°,-1), (360°,0). - **y = cos x**: Wave oscillating between -1 and 1, period 360°. Key points: (0,1), (90°,0), (180°,-1), (270°,0), (360°,1). Cos x is sin x shifted left 90°. - **y = tan x**: Branches repeating every 180°, asymptotes at x = 90°, 270°, etc. Passes through (0,0). ### Solving Trig Equations 1. Use calculator to find the first solution (acute angle). 2. Use graph symmetry to find other solutions in 0° ≤ x ≤ 360°. - **sin x = k**: Solutions are x and 180° - x. - **cos x = k**: Solutions are x and 360° - x. - **tan x = k**: Solutions are x and x + 180°. 3. Always check solutions lie in the given interval. ### Important Relationships - sin(180° - α) = sin α - cos(180° - α) = -cos α - cos(360° + α) = cos α - For obtuse angles: sin > 0, cos < 0. ### Exam Tips - Know how to use sin⁻¹, cos⁻¹, tan⁻¹ on your calculator. - Sketch the graph to visualize solutions. - Rearrange equations first (e.g., 3cos x = 1 → cos x = 1/3).

Period: The interval after which a trigonometric graph repeats. For sin x and cos x, period = 360°. For tan x, period = 180°.
Asymptote: A vertical line that the graph of tan x approaches but never touches, occurring at x = 90°, 270°, etc.
Acute angle: An angle between 0° and 90°.
Obtuse angle: An angle between 90° and 180°.
Inverse trigonometric functions: Functions (sin⁻¹, cos⁻¹, tan⁻¹) used to find the angle from a trigonometric ratio.

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