Trigonometry Identities

Learn the Quotient Identity and Pythagorean Identity

CAPS Grade 10 Mathematics

Trigonometric identities are equations that are true for all angles. The two main identities for Grade 10 are the Quotient Identity and the Pythagorean Identity.

What You'll Learn

Use the Quotient Identity: tan θ = sin θ / cos θ

Use the Pythagorean Identity: sin²θ + cos²θ = 1

Simplify trigonometric expressions

Find unknown trig values using identities

Prove simple trigonometric relationships

Quiz 1: Basic Ratios

What is tan θ equal to

A) opposite/hypotenuse

B) adjacent/hypotenuse

C) opposite/adjacent

D) hypotenuse/opposite

1. The Two Main Identities

Quotient Identity

tan θ = sin θ / cos θ

This identity connects tangent to sine and cosine.

Valid when: cos θ ≠ 0 (θ ≠ 90°, 270°)

Pythagorean Identity

sin²θ + cos²θ = 1

This identity comes from the Pythagorean theorem.

Valid for: All angles θ

Quiz 2: Pythagorean Identity

If sin θ = 0.6, what is sin²θ + cos²θ equal to

A) 0.36

B) 0.64

C) 1

D) 1.36

2. Quotient Identity in Action

Example 1: Using tan θ = sin θ / cos θ

Problem

If sin θ = 0.6 and cos θ = 0.8, find tan θ using the quotient identity.

Step-by-Step Solution

tan θ = sin θ / cos θ
tan θ = 0.6 / 0.8
tan θ = 0.75 or 3/4

Answer: tan θ = 0.75

Example 2: Finding sin θ from tan θ

Problem

If tan θ = 2 and cos θ = 0.4472, find sin θ.

Step-by-Step Solution

tan θ = sin θ / cos θ
2 = sin θ / 0.4472
sin θ = 2 × 0.4472 = 0.8944

Answer: sin θ ≈ 0.8944

Quiz 3: Quotient Identity

If sin θ = 0.8 and cos θ = 0.6, what is tan θ

A) 0.75

B) 1.25

C) 1.33

D) 0.48

3. Pythagorean Identity in Action

Example 3: Finding cos θ from sin θ

Problem

If sin θ = 0.6 and θ is acute (0° to 90°), find cos θ using the Pythagorean identity.

Step-by-Step Solution

sin²θ + cos²θ = 1
(0.6)² + cos²θ = 1
0.36 + cos²θ = 1
cos²θ = 1 - 0.36 = 0.64
cos θ = √0.64 = 0.8 (positive since θ acute)

Answer: cos θ = 0.8

Example 4: Finding sin θ from cos θ

Problem

If cos θ = 12/13 and θ is acute, find sin θ.

Step-by-Step Solution

sin²θ + (12/13)² = 1
sin²θ + 144/169 = 1
sin²θ = 1 - 144/169 = 25/169
sin θ = √(25/169) = 5/13

Answer: sin θ = 5/13

Quiz 4: Pythagorean Identity

If cos θ = 0.8, what is sin θ (θ acute)

A) 0.2

B) 0.36

C) 0.6

D) 0.64

4. Simplifying Expressions

Example 5: Simplify (sin θ / tan θ)

Problem

Simplify: sin θ ÷ tan θ

Step-by-Step Solution

tan θ = sin θ / cos θ
sin θ ÷ tan θ = sin θ ÷ (sin θ / cos θ)
= sin θ × (cos θ / sin θ)
= cos θ

Answer: cos θ

Example 6: Simplify (1 - cos²θ) / sin θ

Problem

Simplify: (1 - cos²θ) / sin θ

Step-by-Step Solution

From Pythagorean identity: 1 - cos²θ = sin²θ
Expression becomes: sin²θ / sin θ
= sin θ

Answer: sin θ (when sin θ ≠ 0)

Quiz 5: Simplifying

Simplify: cos θ × tan θ

A) sin θ

B) cos θ

C) tan θ

D) 1

5. Useful Variations

From Quotient Identity

tan θ = sin θ / cos θ
sin θ = tan θ × cos θ
cos θ = sin θ / tan θ

From Pythagorean Identity

sin²θ = 1 - cos²θ
cos²θ = 1 - sin²θ
sin θ = √(1 - cos²θ) (acute)
cos θ = √(1 - sin²θ) (acute)

Quiz 6: Combined Identities

If sin θ = 3/5 and cos θ = 4/5, what is tan θ

A) 3/4

B) 4/3

C) 3/5

D) 4/5

6. Common Mistakes

Mistake:

tan θ = sin θ / cos θ only when cos θ ≠ 0 (θ ≠ 90°, 270°)

Mistake:

Forgetting the square root when using sin²θ = 1 - cos²θ - always take ±√

Mistake:

sin²θ means (sin θ)², not sin(θ²)

Mistake:

For Grade 10, all angles are acute (0° to 90°), so all ratios are positive

7. Practice Questions

If sin θ = 0.5, find cos θ (θ acute).

Answer: cos θ = √3/2 ≈ 0.866

Simplify: cos θ × tan θ

Answer: sin θ

If tan θ = 1 and θ acute, find sin θ and cos θ.

Answer: sin θ = cos θ = √2/2

8. Summary

Quotient Identity

tan θ = sin θ / cos θ

Connects tangent to sine and cosine

Pythagorean Identity

sin²θ + cos²θ = 1

Comes from the Pythagorean theorem

For Grade 10

θ is acute (0° to 90°)
All ratios are positive
Use identities to simplify and find unknown values

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