Lesson 12 ~25 min Unit 3 · Trigonometry +85 XP

Finding Angles — Mixed Practice

Choose between $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$ based on which two sides are known. Use the third-angle property ($\theta + \phi = 90°$) to find both acute angles.

Today's hook: Three sides known, no angles. Or two sides known, no angles. Either way, inverse trig (plus the angle-sum rule) gives you EVERY angle in the triangle.

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Think First

warm-up

A right triangle has acute angles $\theta$ and $\phi$. What is $\theta + \phi$? (Hint: all three angles sum to 180°.)

Record your answer in your workbook.

The Big Idea

+5 XP

In any right triangle, the two acute angles add to 90° (because all three angles sum to 180° and one of them is 90°). So finding ONE acute angle gives you the other for free.

$\theta + \phi = 90°$ for the two acute angles of a right triangle. If $\theta = 35°$, then $\phi = 55°$. We call these complementary angles. Combined with inverse trig, this means we usually only need ONE inverse-trig computation to find ALL angles in the triangle.

$\theta + \phi = 90°$ for the two acute angles

One inverse, two angles

Compute one acute angle; the other = 90° minus it.

Complementary

The two acute angles ‘complete’ the right angle.

All three sides? Even better

With all three sides you can pick ANY ratio to find an angle.

What You'll Master

objectives

Know

The two acute angles in a right triangle sum to 90°
Choosing the right inverse function based on the side pair
Pythagoras gives the missing side if needed first

Understand

Why complementary angles work
How to find ALL angles in a right triangle from any two sides
When more than one inverse calculation is sensible (and when not)

Can Do

Find both acute angles in any right triangle
Choose the best ratio for a given side pair
Cross-check using complementary angles

Words You Need

vocabulary

Complementary anglesTwo angles that add to 90°. The two acute angles of a right triangle are always complementary.

Angle sum of triangleAll three angles of any triangle add to 180°.

Acute angleAn angle less than 90°. Both non-right angles in a right triangle are acute.

$\phi$ (phi)Another Greek letter used to label an angle, like $\theta$.

Third-angle propertyIn a right triangle, the third angle is $90° - \theta$ if one angle is $\theta$ and another is 90°.

$\sin\theta = \cos\phi$Useful identity when $\theta$ and $\phi$ are complementary: $\sin\theta = \cos(90°-\theta)$.

Spot the Trap

heads-up

Wrong: Forgetting that the second acute angle is automatically 90° minus the first — doing redundant calculator work.

Right: Find one acute angle with inverse trig; the other = $90° - \theta$.

Wrong: Using the same side as both opp and adj for the two different acute angles.

Right: Opp and adj SWAP when you switch reference angle — re-label before doing the second angle.

Two Angles, One Triangle

+5 XP

When solving for both acute angles, you only need ONE inverse-trig call. The other comes from the angle sum.

Given sides 3, 4, 5 with right angle, find both acute angles. For the angle opposite side 3: $\sin\theta = 3/5$, $\theta = \sin^{-1}(0.6) \approx 36.87°$. For the other acute angle: $\phi = 90° - 36.87° \approx 53.13°$. Check: $36.87 + 53.13 = 90$ ✓

Compute one angle → subtract from 90° for the other

Subtract, don't recompute

$\phi = 90° - \theta$ is faster than another inverse trig call.

Sanity check

Adding the two acute angles must give 90°.

Identity bonus

$\sin\theta = \cos\phi$ when they're complementary.

All Three Sides? Pick Smart

+5 XP

When you know all three sides, you can use any of the three ratios. Pick the one that avoids rounding errors.

Situation	Best ratio
Hyp is ‘clean’ (e.g. 10 or 13)	Use sin or cos
Hyp is messy (irrational)	Use tan with the two legs
Always avoid using a side you computed yourself	Use given sides only when possible

All three sides known → choose the cleanest pair

Use exact sides

Avoid using a side you computed — rounding errors stack up.

Tan is reliable

If both legs are given, tan never involves the (possibly irrational) hypotenuse.

Three checks

You can verify using all three ratios — consistency is reassuring.

Watch Me Solve It · 3 examples

Watch Me Solve It · Both acute angles from 3-4-5

+15 XP per step

PROBLEM

In a 3-4-5 right triangle, find both acute angles.

1

First angle

Opposite side 3: $\sin\theta = 3/5$, $\theta = \sin^{-1}(0.6) \approx 36.87°$
2

Second angle

$\phi = 90° - 36.87° = 53.13°$
3

Check

$36.87° + 53.13° = 90°$ ✓

Answer$\theta \approx 36.87°$, $\phi \approx 53.13°$

Watch Me Solve It · Find an angle in a tower problem

+15 XP per step

PROBLEM

A flagpole 9 m tall casts a shadow 12 m long. At what angle above horizontal is the sun (1 d.p.)?

1

Set up

opp = 9 (height), adj = 12 (shadow). Use tan.
2

Compute

$\tan\theta = 9/12 = 0.75$
3

Inverse

$\theta = \tan^{-1}(0.75) \approx 36.9°$

Answer$\theta \approx 36.9°$

Watch Me Solve It · Angles of a 5-12-13 triangle

+15 XP per step

PROBLEM

Find both acute angles of a 5-12-13 right triangle (1 d.p.).

1

First

Use sin with opp = 5, hyp = 13: $\sin\theta = 5/13 \approx 0.3846$
2

Inverse

$\theta \approx 22.6°$
3

Second

$\phi = 90° - 22.6° = 67.4°$

Answer$\theta \approx 22.6°$, $\phi \approx 67.4°$

Common Pitfalls

heads-up

Re-computing the second angle from scratch

Doing two inverse-trig calculations when subtraction would do.

Fix: $\phi = 90° - \theta$. One inverse trig is enough.

Same labelling for both angles

Forgetting that opp and adj SWAP when changing reference angle.

Fix: Re-label opp/adj before computing the second angle's ratio.

Using rounded sides

Using a previously rounded answer in another calculation, amplifying errors.

Fix: Use given (unrounded) sides whenever possible.

Copy Into Your Books

Sum rule

$\theta + \phi = 90°$
Complementary
Find one, subtract

All three sides

Any ratio works
Avoid computed sides
Use exact pairs

Same method

Label sides
Pick ratio
Inverse trig

Sanity check

Angles sum to 90°
Acute < 90°
Greater angle opposite longer leg

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Angle Combo

4 problems

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

1 One acute angle is 30°. The other is?

$90° - 30° = 60°$.60°
2 In 5-12-13: find angle opposite side 5.

$\sin^{-1}(5/13) \approx 22.6°$.$\approx 22.6°$
3 In 5-12-13: find the OTHER acute angle.

$90 - 22.6 = 67.4°$.$\approx 67.4°$
4 Both legs 5. Find an acute angle.

$\tan\theta = 5/5 = 1$, $\theta = 45°$.45°

Complete in your workbook.

Quick Check · 5 questions

In a right triangle, one acute angle is 25°. The other is:

+10 XP

In a 7-24-25 right triangle, the smallest acute angle is (1 d.p.):

+10 XP

The two acute angles of a right triangle are called:

+10 XP

Legs both 4 cm. An acute angle is:

+10 XP

opp = 6, hyp = 10. The angle $\theta$ is (1 d.p.):

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. In a right triangle, opp = 9 and adj = 12 (relative to $\theta$). (a) Find $\theta$ (1 d.p.). (b) Find the other acute angle $\phi$. (c) Verify $\theta + \phi = 90°$.

Answer in your workbook.

ApplyEasy2 MARKS

Q7. A flagpole 8 m tall casts a shadow 10 m long. At what angle (1 d.p.) does the sun strike the ground?

Answer in your workbook.

ReasonHard4 MARKS

Q8. A right triangle has hypotenuse 17 cm and one leg 8 cm. (a) Find the other leg. (b) Find both acute angles (1 d.p.). (c) Show that the two acute angles add to 90°.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $90 - 25 = 65$.

2. A — $\sin^{-1}(7/25) \approx 16.3°$.

3. C — Sum to 90°.

4. D — $\tan^{-1}(1) = 45°$.

5. A — $\sin^{-1}(0.6) \approx 36.9°$.

Show Your Working Model Answers

Q6 (3 marks): (a) $\tan\theta = 9/12 = 0.75$, $\theta = \tan^{-1}(0.75) \approx 36.9°$ [1]. (b) $\phi = 90° - 36.9° = 53.1°$ [1]. (c) $36.9 + 53.1 = 90$ ✓ [1].

Q7 (2 marks): $\tan\theta = 8/10 = 0.8$ [1]. $\theta = \tan^{-1}(0.8) \approx 38.7°$ [1].

Q8 (4 marks): (a) Other leg $= \sqrt{17^2-8^2}=\sqrt{225}=15$ cm [1]. (b) $\theta = \sin^{-1}(8/17) \approx 28.1°$ (angle opposite the 8) [1]. $\phi = \sin^{-1}(15/17) \approx 61.9°$ (angle opposite the 15) [1]. (c) $28.1 + 61.9 = 90.0°$ ✓ (both acute angles complementary as expected) [1].

Stretch Challenge · +25 XP, +10 coins

The 30-60-90 triangle

A right triangle has legs of length 1 and $\sqrt{3}$, and hypotenuse 2. Find both acute angles exactly. What's special about this triangle?

Reveal solution

$\tan\theta = 1/\sqrt{3}$, so $\theta = 30°$ exactly. The other angle $= 90° - 30° = 60°$. This is the famous 30-60-90 triangle, half of an equilateral triangle — its exact-value ratios are needed for advanced trig.

Quick Review

Sum rule

$\theta + \phi = 90°$

One inverse

Find one acute, subtract for the other

Same ratio rules

opp+hyp→sin, etc.

Complementary

Two acute angles complete the 90°

All three sides

Pick cleanest pair

Verify

Angles should sum to 90°

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