S3 Mathematics - Term 1

Topic 1: Trigonometry I

Lesson 5: Finding Angles from Ratios

Duration: 40 minutes

Learning Outcomes

By the end of this lesson, you should be able to:

• Understand inverse trigonometric functions
• Use calculator to find angles from trig ratios
• Apply sin⁻¹, cos⁻¹, tan⁻¹ to triangle problems

Review: The Forward Problem

Question: If angle = 30 degrees, what is sin 30 degrees?

Answer: 0.5 (using calculator)

Today: We work BACKWARDS!

The Inverse Problem

Question: If sin theta = 0.5, what is theta?

We need a way to work backwards from the ratio to the angle!

Inverse Functions

Just like other operations have inverses:

• Addition Subtraction
• Multiplication Division
• Sine Inverse Sine

Notation

Inverse trig functions:

• sin⁻¹ or arcsin (inverse sine)
• cos⁻¹ or arccos (inverse cosine)
• tan⁻¹ or arctan (inverse tangent)

IMPORTANT: The "-1" is NOT an exponent!
It means "inverse"

What sin⁻¹ Means

sin⁻¹(0.5) means:

"Find the angle whose sine is 0.5"

Answer: 30 degrees

Forward and Inverse

Finding Inverse Buttons

On your calculator:

• Look above sin, cos, tan buttons
• May need to press SHIFT, 2nd, or INV first
• Look for: sin⁻¹, cos⁻¹, tan⁻¹

Let's Try: sin⁻¹(0.5)

Steps:

1. Press SHIFT or 2nd (if needed)
2. Press sin (to get sin⁻¹)
3. Enter 0.5
4. Press = or EXE

Answer: 30 degrees

Practice Together

Find these angles:

1. cos⁻¹(0.5)
2. tan⁻¹(1)

Answers:

1. 60 degrees
2. 45 degrees

More Examples

• sin⁻¹(0.707) ≈ 45 degrees
• cos⁻¹(0.866) ≈ 30 degrees
• tan⁻¹(1.732) ≈ 60 degrees

Remember: tan values can be > 1, this is normal!

Finding Unknown Angles

Example 1: Using sin⁻¹

Triangle: opposite = 6 cm, hypotenuse = 10 cm

Step 1: sin theta = 6/10 = 0.6

Step 2: theta = sin⁻¹(0.6)

Step 3: theta ≈ 36.9 degrees

Example 2: Using cos⁻¹

Triangle: adjacent = 12 cm, hypotenuse = 15 cm

Step 1: cos theta = 12/15 = 0.8

Step 2: theta = cos⁻¹(0.8)

Step 3: theta ≈ 36.9 degrees

Example 3: Using tan⁻¹

Triangle: opposite = 8 cm, adjacent = 6 cm

Step 1: tan theta = 8/6 = 1.333

Step 2: theta = tan⁻¹(1.333)

Step 3: theta ≈ 53.1 degrees

Pair Practice 1

Triangle: opposite = 7 cm, hypotenuse = 25 cm

Find the angle theta

Answer 1

sin theta = 7/25 = 0.28

theta = sin⁻¹(0.28)

theta ≈ 16.3 degrees

Pair Practice 2

Triangle: adjacent = 5 cm, opposite = 12 cm

Find the angle theta

Answer 2

tan theta = 12/5 = 2.4

theta = tan⁻¹(2.4)

theta ≈ 67.4 degrees

Verification Method

After finding an angle, check your answer!

Found theta = 36.9 degrees?
Check: sin 36.9 degrees should give ≈ 0.6 ✓

Great way to verify!

Summary

The Complete Picture

Forward: angle → ratio
sin 30 degrees = 0.5

Inverse: ratio → angle
sin⁻¹(0.5) = 30 degrees

Both directions are useful!

When to Use Each

Know angle, want sides?
Use sin, cos, tan

Know sides, want angle?
Use sin⁻¹, cos⁻¹, tan⁻¹

Exit Questions

Find angle theta (round to 1 decimal place):

1. sin theta = 0.4
2. cos theta = 0.75
3. Triangle: opposite = 9, adjacent = 12

Answers

1. theta = sin⁻¹(0.4) ≈ 23.6 degrees

2. theta = cos⁻¹(0.75) ≈ 41.4 degrees

3. tan theta = 9/12 = 0.75
   theta = tan⁻¹(0.75) ≈ 36.9 degrees

Homework

1. Find angles: sin theta = 0.3, cos theta = 0.6, tan theta = 1.5

2. Triangle problems with given sides

3. Challenge: In a right triangle, one angle is 35 degrees. What is the other non-right angle?

Next Lesson

Lesson 6: Solving Right Triangles

• Find ALL missing sides and angles
• Use both forward and inverse together
• Complete triangle problems!

Credits

Created: December 2025
Based on: NCDC Lower Secondary Mathematics Syllabus (2019)

Source: National Curriculum Development Centre (NCDC), Uganda