To find an unknown angle, work out the ratio from the two known sides, then apply the inverse trig function. The calculator gives decimal degrees — convert to degrees and minutes for exam answers.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
If $\sin\theta = 0.5$, you probably know that $\theta = 30°$. But what if $\sin\theta = 0.73$? What operation would you use on your calculator? And what does the answer mean — is it always the only possible angle?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Come back to this at the end of the lesson.
Wrong: Rounding 4.5 down gives 4.
Right: Standard rounding rounds 4.5 up to 5; rounding to the nearest even number (banker's rounding) is different.
Finding Angles with Inverse Trig
Always check your units before substituting into formulas. Converting to consistent units is a common source of errors in assessment tasks.
When you know a trig ratio (e.g. $\sin\theta = 0.6$) but not the angle, press the inverse trig function on your calculator. $\sin^{-1}(0.6)$ gives the angle whose sine is 0.6.
A right-angled triangle has opposite side 7 cm and hypotenuse 11 cm. Find the angle $\theta$ opposite the 7 cm side, in degrees and minutes.
A building is 45 m tall. An observer stands 60 m from the base. Find the angle of elevation to the top of the building, in degrees and minutes.
A right-angled triangle has legs 8 m and 15 m. Find both acute angles in degrees and minutes.
Section A — Finding a Single Angle
Section B — Finding Both Angles
Section C — Practical Applications
$\sin^{-1}(5/13) \approx 22.62° \approx \mathbf{22°37'}$
$\cos^{-1}(9/15) = \cos^{-1}(0.6) \approx 53.13° \approx \mathbf{53°8'}$
$\tan^{-1}(7/24) \approx 16.26° \approx \mathbf{16°16'}$
$\sin^{-1}(4.2/9.8) \approx \sin^{-1}(0.4286) \approx 25.38° \approx \mathbf{25°23'}$
$\cos^{-1}(8.5/14) \approx \cos^{-1}(0.6071) \approx 52.61° \approx \mathbf{52°37'}$
$\alpha = \tan^{-1}(5/12) \approx 22°37'$; $\beta = 90° - 22°37' = \mathbf{67°23'}$
One leg $= 7$, H $= 25$; $\alpha = \sin^{-1}(7/25) \approx 16°16'$; $\beta = 90° - 16°16' = \mathbf{73°44'}$
$\tan^{-1}(1.5/9) \approx \tan^{-1}(0.1\overline{6}) \approx 9.46° \approx \mathbf{9°28'}$
$\tan^{-1}(30/40) = \tan^{-1}(0.75) \approx 36.87° \approx \mathbf{36°52'}$ east of north
$\tan^{-1}(48/120) = \tan^{-1}(0.4) \approx 21.80° \approx \mathbf{21°48'}$
$\tan^{-1}(280/650) \approx \tan^{-1}(0.4308) \approx 23.32° \approx \mathbf{23°19'}$
$\tan^{-1}(12/5) \approx 67.38° \approx \mathbf{67°23'}$
Car A: $d = 80/\tan32° \approx \mathbf{128.0 \text{ m}}$; Car B: $d = 80/\tan18° \approx \mathbf{246.2 \text{ m}}$
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
Multiple Choice
1 In a right-angled triangle, the side opposite angle $\theta$ is 9 m and the hypotenuse is 14 m. Angle $\theta$, in degrees and minutes, is closest to:
? Regarding this topic, 1 In a right-angled triangle, the side opposite angle $\theta$ is 9 m and the hypotenuse is 14 m. Angle $\theta$, in degrees and minutes, is closest to:
2 A 10-metre ladder leans against a vertical wall. The foot of the ladder is 4.8 m from the base of the wall. The angle the ladder makes with the ground, in degrees and minutes, is closest to:
? Regarding this topic, 2 A 10-metre ladder leans against a vertical wall. The foot of the ladder is 4.8 m from the base of the wall. The angle the ladder makes with the ground, in degrees and minutes, is closest to:
3 A vertical cliff is 75 m high. From the base, the angle of elevation to the top of a tree on top of the cliff is 42°. The horizontal distance from the base of the cliff to the observer is 85 m. The angle of elevation to the bottom of the cliff is:
? Regarding this topic, 3 A vertical cliff is 75 m high. From the base, the angle of elevation to the top of a tree on top of the cliff is 42°. The horizontal distance from the base of the cliff to the observer is 85 m. The angle of elevation to the bottom of the cliff is:
Short Answer
SA 4 3 marks A right-angled triangle has a hypotenuse of 20 cm and one side of 12 cm.
(a) Find the angle between the 12 cm side and the hypotenuse, in degrees and minutes. (2 marks)
(b) Find the other acute angle. (1 mark)
12 cm is adjacent to the angle (angle between it and H); $\cos\theta = 12/20 = 0.6$; $\theta = \cos^{-1}(0.6) \approx 53.13° \approx \mathbf{53°8'}$
$90° - 53°8' = \mathbf{36°52'}$
SA 5 3 marks A pilot is flying at an altitude of 1500 m. She spots a runway that is 4200 m horizontally from directly below the aircraft.
(a) Draw and label the right-angled triangle. (1 mark)
(b) Find the angle of depression from the aircraft to the runway, in degrees and minutes. (2 marks)
Right angle at the point directly below the aircraft; altitude $= 1500$ m (O); horizontal distance $= 4200$ m (A); angle of depression at aircraft position
$\tan\theta = 1500/4200 = 0.3571$; $\theta = \tan^{-1}(0.3571) \approx 19.65° \approx \mathbf{19°39'}$
SA 6 4 marks A 15-metre telephone pole stands vertically. A support wire is attached from the top of the pole to a point on the ground. Safety regulations require the angle between the wire and the ground to be between 55° and 70°.
(a) If the wire is anchored 9 m from the base of the pole, find the angle the wire makes with the ground (in degrees and minutes). (2 marks)
(b) Does this satisfy the safety regulation? Justify your answer. (1 mark)
(c) Find the minimum distance from the base of the pole at which the wire can be anchored while still satisfying the regulation. Give your answer to 1 decimal place. (1 mark)
$\tan\theta = 15/9 = 1.\overline{6}$; $\theta = \tan^{-1}(5/3) \approx 59.04° \approx \mathbf{59°2'}$
$55° \leq 59°2' \leq 70°$ — Yes, the angle satisfies the regulation.
Max angle is 70°; $\tan70° = 15/d_{\min}$; $d_{\min} = 15/\tan70° \approx 15/2.747 \approx \mathbf{5.5 \text{ m}}$
Right-Angled Trigonometry: Finding Unknown Angles