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Right-Angled Trigonometry

Select the skills to practice, and then click Go!

Foundational skills
Identify the hypotenuse
\[ \text{Opposite the right angle} \]
Find the longest side, opposite the right angle.
Identify opposite and adjacent
\[ \text{Opposite or adjacent to } \theta \]
Name the side opposite or next to a marked angle.
Choose sin, cos or tan
\[ \text{SOH} \quad \text{CAH} \quad \text{TOA} \]
Decide which ratio links the sides you know and want.
Write a trig ratio as a fraction
\[ \cos\theta = \frac{12}{13} \]
Write sin, cos or tan of an angle as a fraction of two sides.
Use a calculator: sin, cos and tan
\[ \tan 38^\circ = 0.781 \]
Find sin, cos or tan of an angle using a calculator.
Use a calculator: inverse trig
\[ \cos x = 0.42 \Rightarrow x = 65.2^\circ \]
Use sin⁻¹, cos⁻¹ or tan⁻¹ to find an angle.
Solve: unknown on top
\[ \frac{x}{9} = \sin 35^\circ \]
Solve a trig equation where x is on top of the fraction.
Solve: unknown on the bottom
\[ \frac{8}{x} = \cos 50^\circ \]
Solve a trig equation where x is on the bottom of the fraction.
Finding a side: unknown on top
Opposite, given the hypotenuse
\[ x = 12 \times \sin 35^\circ \]
Use sin to find the opposite side.
Adjacent, given the hypotenuse
\[ x = 12 \times \cos 35^\circ \]
Use cos to find the adjacent side.
Opposite, given the adjacent
\[ x = 10 \times \tan 40^\circ \]
Use tan to find the opposite side.
Finding a side: unknown on the bottom
Hypotenuse, given the opposite
\[ x = \frac{8}{\sin 35^\circ} \]
Use sin to find the hypotenuse.
Hypotenuse, given the adjacent
\[ x = \frac{8}{\cos 35^\circ} \]
Use cos to find the hypotenuse.
Adjacent, given the opposite
\[ x = \frac{9}{\tan 40^\circ} \]
Use tan to find the adjacent side.
Finding an angle
Angle using sin⁻¹
\[ \theta = \sin^{-1}\left(\frac{7}{12}\right) \]
Find an angle from the opposite and the hypotenuse.
Angle using cos⁻¹
\[ \theta = \cos^{-1}\left(\frac{5}{8}\right) \]
Find an angle from the adjacent and the hypotenuse.
Angle using tan⁻¹
\[ \theta = \tan^{-1}\left(\frac{9}{4}\right) \]
Find an angle from the opposite and the adjacent.
Exact values
Recall exact values
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Know sin, cos and tan of 0°, 30°, 45°, 60° and 90°.
Angle from an exact value
\[ \tan x = 1 \Rightarrow x = 45^\circ \]
Find the angle that gives an exact trig value.
Find a side (non-calculator)
\[ \frac{x}{14} = \frac{1}{2} \Rightarrow x = 7 \]
Use exact values to find a side without a calculator.
Find an angle (non-calculator)
\[ \sin\theta = \frac{5}{10} \Rightarrow 30^\circ \]
Use exact values to find an angle without a calculator.
Angles of elevation and depression
Identify the angle
\[ \text{Measure from the horizontal} \]
Spot angles of elevation and depression on a diagram.
Find a height (elevation)
\[ h = 25 \times \tan 32^\circ \]
Use an angle of elevation to find a height.
Find a distance (depression)
\[ d = \frac{60}{\tan 25^\circ} \]
Use an angle of depression to find a distance.
Find the angle
\[ \theta = \tan^{-1}\left(\frac{40}{65}\right) \]
Find an angle of elevation or depression.
Extensions
Pythagoras or trigonometry?
\[ \text{Is an angle involved?} \]
Decide whether to use Pythagoras or trig, then find x.
Two triangles sharing a side
\[ \text{Find the shared side first} \]
Find the shared side first, then use it in the second triangle.
Isosceles triangles
\[ \text{Split into two right triangles} \]
Split an isosceles triangle into two right-angled triangles.
Area using trigonometry
\[ \text{Area} = \tfrac{1}{2} \times b \times h \]
Use trig to find the base and height, then the area.
Bearings with right-angled trig
\[ 052^\circ \text{ for } 30\text{km} \]
Find how far north or east something has travelled.
Timer (Optional)
Question