Right Angled Triangles

Pythagoras' Theorem

• Pythagoras' theorem relates the sides of a right-angled triangle: $a^{2} + b^{2} = c^{2}$.
• c is the hypotenuse (longest side, opposite the right angle).
• To find the hypotenuse: $c = \sqrt{a^{2} + b^{2}}$.
• To find a shorter side: $a = \sqrt{c^{2} - b^{2}}$.

In a right-angled triangle, the sides are 3 cm and 4 cm. What is the length of the hypotenuse?

• 5 cm
• 6 cm
• 7 cm
• 25 cm

Finding a Shorter Side

• To find a shorter side, subtract the squares: $a = \sqrt{c^{2} - b^{2}}$.
• Always subtract the smaller square from the larger one.
• If the hypotenuse is 13 cm and one side is 5 cm, the other side $is \sqrt{13^{2} - 5^{2}} = \sqrt{169 - 25} = \sqrt{144} = 12$ cm.

In a right-angled triangle with hypotenuse 10 cm and one side 6 cm, the other side is ____ cm.

SOHCAHTOA

• Trigonometry uses ratios of sides in right-angled triangles.
• sin θ = Opposite / Hypotenuse (SOH).
• cos θ = Adjacent / Hypotenuse (CAH).
• tan θ = Opposite / Adjacent (TOA).

Which ratio would you use to find the angle if you know the opposite and adjacent sides?

• sin
• cos
• tan
• Pythagoras

Finding Missing Lengths with SOHCAHTOA

• Label the sides: H (hypotenuse), O (opposite), A (adjacent) relative to the given angle.
• Identify which two sides you have and use the correct ratio.
• Substitute and solve: e.g., if you have angle and adjacent, use cos to find hypotenuse.

In a right-angled triangle with angle 30° and adjacent side 10 cm, the hypotenuse is ____ cm (to 3 s.f.).

Finding Missing Angles with SOHCAHTOA

• Use inverse trig functions: sin⁻¹, cos⁻¹, tan⁻¹.
• If tan θ $= \frac{3}{4}$, then θ = tan⁻¹(3/4).
• Round angles to 1 decimal place unless told otherwise.

In a right-angled triangle, opposite $= 5$ cm, hypotenuse $= 13$ cm. What is the angle θ?

• sin⁻¹(5/13)
• cos⁻¹(5/13)
• tan⁻¹(5/13)
• sin⁻¹(13/5)

Angles of Elevation and Depression

• Angle of elevation: measured upwards from horizontal.
• Angle of depression: measured downwards from horizontal.
• These angles are equal to alternate angles in diagrams.
• Use tan often to find heights or distances.

Which of these are true about angles of elevation and depression?

• Angle of elevation is measured from the horizontal looking up.
• Angle of depression is measured from the vertical looking down.
• Angles of elevation and depression are always equal.
• They can be used with right-angled trigonometry.

Exact Trig Values

• Know exact values for 0°, 30°, 45°, 60°, 90°.
• $\sin 30^{\circ} = \frac{1}{2}, \sin 45^{\circ} = \sqrt{2}/2, \sin 60^{\circ} = \sqrt{3}/2$.
• $\cos 30^{\circ} = \sqrt{3}/2, \cos 45^{\circ} = \sqrt{2}/2, \cos 60^{\circ} = \frac{1}{2}$.
• $\tan 30^{\circ} = \sqrt{3}/3, \tan 45^{\circ} = 1, \tan 60^{\circ} = \sqrt{3}$.

The exact value of sin 60° is ____.

Using Exact Trig Values

• In non-calculator questions, substitute exact values.
• For example, if $\cos 45^{\circ} = \frac{x}{12}$, then $x = 12 \times \sqrt{2}/2 = 6\sqrt{2}$.
• Sketch the special triangles (30-60-90 and 45-45-90) to help.

If $\tan 30^{\circ} = \frac{x}{6}$, what is x?

• $2\sqrt{3}$
• $\sqrt{3}$
• $3\sqrt{3}$
• $6\sqrt{3}$

Problem Solving with Right-Angled Triangles

• Combine Pythagoras and trigonometry as needed.
• Draw diagrams and label all given information.
• Break complex shapes into right-angled triangles.
• Leave answers in exact form until the final step.

Which strategies are helpful for solving right-angled triangle problems?

• Always round intermediate steps.
• Label sides H, O, A relative to the angle.
• Use Pythagoras only for finding the hypotenuse.
• Check your calculator is in degree mode.