Lesson 11 ~25 min Unit 3 · Trigonometry +85 XP

Finding Angles with Inverse Trig

When you know two sides but not the angle, use $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$ — the inverse trig functions — to recover $\theta$.

Today's hook: You know the opposite (4 m) and the hypotenuse (7 m). You want the ANGLE between them. How do you UN-do a sine to find $\theta$?

0/5QUESTS

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Think First

warm-up

$\sin\theta = 0.5$. What is $\theta$? Hint: at which classic angle does sine equal exactly one-half?

Record your answer in your workbook.

The Big Idea

+5 XP

To find an angle when you know its sine/cosine/tangent, use the inverse functions $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$ (also written arcsin, arccos, arctan). They are the ‘undo’ buttons.

If $\sin\theta = x$ then $\theta = \sin^{-1}(x)$. Same for cos and tan. On the calculator: SHIFT (or 2nd) then the trig key. The inverse takes a ratio (0 to 1 for sin/cos) and returns the corresponding angle.

$\theta = \sin^{-1}(\text{opp}/\text{hyp})$ etc.

Set up the ratio first

Always write the equation $\sin\theta = $... before applying the inverse.

SHIFT-sin

On the calculator: SHIFT (2nd) followed by sin to get $\sin^{-1}$.

Acute answer

For right-triangle problems, $\theta$ is between 0° and 90°.

What You'll Master

objectives

Know

The inverse functions $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$
How to set up a ratio then apply the inverse
That inverse trig gives an acute angle in right triangles

Understand

Why we need the inverse to find an angle
That $\sin^{-1}$ is NOT the same as $1/\sin$
That for acute angles the inverse always gives 0° to 90°

Can Do

Find an angle given any two sides
Round angles to the nearest degree or one decimal place
Use SHIFT (or 2nd) on the calculator correctly

Words You Need

vocabulary

$\sin^{-1}(x)$ / arcsinThe angle whose sine is $x$. Reads ‘sine inverse of $x$’.

$\cos^{-1}(x)$ / arccosThe angle whose cosine is $x$.

$\tan^{-1}(x)$ / arctanThe angle whose tangent is $x$.

SHIFT keyThe calculator modifier that toggles to the inverse trig functions.

Inverse functionA function that undoes another. $\sin^{-1}$ undoes $\sin$.

Acute answerIn right triangles, the inverse always returns an angle between 0° and 90°.

Spot the Trap

heads-up

Wrong: “$\sin^{-1}(x) = 1/\sin(x)$.” Wrong — that's the reciprocal, not the inverse.

Right: $\sin^{-1}$ is the INVERSE FUNCTION — the angle whose sine is $x$.

Wrong: Pressing sin instead of SHIFT-sin — gives a ratio, not an angle.

Right: To find an angle, ALWAYS use SHIFT (or 2nd) before the trig key.

Calculator Steps

+5 XP

Practise the exact key sequence to compute inverse trig without errors.

Goal	Casio FX-82AU keys
$\sin^{-1}(0.5)$	SHIFT → sin → 0.5 → = → 30°
$\cos^{-1}(4/5)$	SHIFT → cos → ( 4 $\div$ 5 ) → = → 36.87°
$\tan^{-1}(2)$	SHIFT → tan → 2 → = → 63.43°

SHIFT + trig key + value + = → angle

DEG mode still

Inverse trig also needs DEG mode for degree answers.

Use exact ratio

Type the full fraction (4 $\div$ 5) rather than rounding first.

Check size

For acute triangles, $\theta$ between 0° and 90°.

Three-Step Angle Method

+5 XP

Finding an angle follows the same pattern as finding a side, just one extra inverse-trig step at the end.

Step	Action
1	Label sides relative to $\theta$
2	Identify which ratio (sin/cos/tan) links the two known sides
3	Set up & apply inverse: $\theta = $ ratio$^{-1}$(value)

Label → pick ratio → apply inverse

Same ratio rules

opp+hyp → sin; adj+hyp → cos; opp+adj → tan.

Compute decimal first

$\theta = \sin^{-1}(0.667)$ — compute the ratio then apply inverse.

Final unit: degrees

Always state the angle in degrees with the ° symbol.

Watch Me Solve It · 3 examples

Watch Me Solve It · Angle from opp and hyp

+15 XP per step

PROBLEM

In a right triangle, opp = 4 m and hyp = 7 m. Find the angle $\theta$ (1 d.p.).

1

Set up ratio

$\sin\theta = 4/7 \approx 0.5714$
2

Apply inverse

$\theta = \sin^{-1}(4/7)$
3

Compute

$\theta \approx 34.8°$

Answer$\theta \approx 34.8°$

Watch Me Solve It · Classic 3-4-5

+15 XP per step

PROBLEM

A right triangle has opp = 3, adj = 4 (relative to $\theta$). Find $\theta$.

1

Use tan

$\tan\theta = 3/4 = 0.75$
2

Inverse

$\theta = \tan^{-1}(0.75)$
3

Compute

$\theta \approx 36.87°$ — the famous 3-4-5 acute angle.

Answer$\theta \approx 36.87°$ (or $\approx 36.9°$)

Watch Me Solve It · Cliff angle of view

+15 XP per step

PROBLEM

You stand 50 m from the base of a cliff that's 30 m tall. At what angle above horizontal do you see the top (1 d.p.)?

1

Set up

adj = 50 (horizontal), opp = 30 (height). Use tan.
2

Apply

$\tan\theta = 30/50 = 0.6$
3

Inverse

$\theta = \tan^{-1}(0.6) \approx 31.0°$

Answer$\theta \approx 31.0°$

Common Pitfalls

heads-up

Pressing sin instead of $\sin^{-1}$

Returns a ratio, not an angle.

Fix: Always SHIFT (or 2nd) first when finding an angle.

Confusing $\sin^{-1}$ with $1/\sin$

Two completely different operations.

Fix: $\sin^{-1}$ is the INVERSE FUNCTION, NOT the reciprocal.

Wrong ratio choice

Same problem as side-finding: picking sin when tan was needed.

Fix: List the two known sides and apply the SOH-CAH-TOA rules.

Copy Into Your Books

Inverse functions

$\sin^{-1}(x)$
$\cos^{-1}(x)$
$\tan^{-1}(x)$

Calculator

SHIFT (2nd) before sin/cos/tan
DEG mode
Returns the angle

Method

Label sides
Pick ratio
Inverse for $\theta$

Sanity

Acute → $0 < \theta < 90°$
$\sin^{-1}(0.5) = 30°$
$\tan^{-1}(1) = 45°$

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Inverse Trig

4 problems

Four quick drills to lock in today's skill. Try each, then reveal the answer.

1 $\sin\theta = 0.5$. Find $\theta$.

$\theta = \sin^{-1}(0.5)$.$\theta = 30°$
2 $\cos\theta = 0.8$. Find $\theta$ (1 d.p.).

$\theta = \cos^{-1}(0.8)$.$\theta \approx 36.9°$
3 $\tan\theta = 1$. Find $\theta$.

$\theta = \tan^{-1}(1)$.$\theta = 45°$
4 opp = 5, hyp = 13. Find $\theta$ (1 d.p.).

$\theta = \sin^{-1}(5/13)$.$\theta \approx 22.6°$

Complete in your workbook.

Quick Check · 5 questions

$\sin\theta = 0.5$. The angle is:

+10 XP

To find $\theta$ when you know $\tan\theta$, press:

+10 XP

opp = 7, hyp = 10. Find $\theta$ (1 d.p.):

+10 XP

$\sin^{-1}(x)$ means:

+10 XP

adj = 6, opp = 6. Find $\theta$:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Find $\theta$ to 1 d.p. in each case. (a) opp = 5, hyp = 13. (b) adj = 8, opp = 6. (c) adj = 4, hyp = 5.

Answer in your workbook.

ApplyEasy2 MARKS

Q7. You stand 20 m from the foot of a 15 m flagpole. At what angle above horizontal do you see the top (1 d.p.)?

Answer in your workbook.

ReasonHard4 MARKS

Q8. A 7 m ladder leans against a wall with its base 2 m from the wall. (a) Find the angle the ladder makes with the ground. (b) Building safety guidance says ladders should be at 70-80° for safety. Is this ladder safe? Explain.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $\sin 30° = 0.5$.

2. D — SHIFT + tan.

3. C — $\sin^{-1}(0.7) \approx 44.4°$.

4. A — Inverse function.

5. A — $\tan^{-1}(1) = 45°$.

Show Your Working Model Answers

Q6 (3 marks): (a) $\theta = \sin^{-1}(5/13) \approx 22.6°$ [1]. (b) $\theta = \tan^{-1}(6/8) \approx 36.9°$ [1]. (c) $\theta = \cos^{-1}(4/5) \approx 36.9°$ [1].

Q7 (2 marks): $\tan\theta = 15/20 = 0.75$ [1]. $\theta = \tan^{-1}(0.75) \approx 36.9°$ [1].

Q8 (4 marks): (a) $\cos\theta = 2/7 \approx 0.2857$ [1]. $\theta = \cos^{-1}(2/7) \approx 73.4°$ [1]. (b) $73.4°$ lies within 70-80°, so the ladder IS safely placed [1]. Explanation: the angle is steep enough to keep the foot from slipping but not too steep to topple [1].

Stretch Challenge · +25 XP, +10 coins

Match the angle to the world

A wheelchair ramp must rise 1 unit for every 12 horizontal units (a 1-in-12 gradient). What angle does this ramp make with the ground (1 d.p.)? Compare with a steep mountain pass that climbs 1 unit in 4 horizontal units.

Reveal solution

Wheelchair: $\theta = \tan^{-1}(1/12) \approx 4.8°$. Mountain: $\theta = \tan^{-1}(1/4) \approx 14.0°$ — nearly three times as steep.

Quick Review

Inverse functions

$\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$

On calculator

SHIFT + trig key

Same ratio rule

opp+hyp → sin, etc.

$\sin^{-1} \neq 1/\sin$

Inverse function, not reciprocal

Acute answer

$0 < \theta < 90°$

Three steps

Label, pick, inverse

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