← Unit 4 Trigonometric Ratios

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Trigonometric Ratios

⏱ 25 min📚 Year 9📈
Think First

If $sin \theta = 3/5$, what is $cos \theta$?

💡 Revision: Ensure you can label sides and use SOH CAH TOA (Lesson 1).

Learning Intentions

Know

  • $sin^2 \theta + cos^2 \theta = 1$
  • $\tan \theta = sin \theta / cos \theta$
  • Complementary angles

Understand

  • Why the identity $sin^2 + cos^2 = 1$ holds
  • How trig ratios change as the angle increases

Can Do

  • Use the Pythagorean identity
  • Find all three ratios given one
  • Solve problems involving complementary angles
IdentityComplementaryPythagoreanRatio
Learn Phase
1

The Pythagorean Identity

A fundamental relationship

For any angle $\theta$:

Pythagorean Identity

$sin^2 \theta + cos^2 \theta = 1$

This follows from Pythagoras: if the hypotenuse is 1, then $\text{opp}^2 + \text{adj}^2 = 1$.

Example: If $sin \theta = 3/5$, then:

$cos^2 \theta = 1 - (3/5)^2 = 1 - 9/25 = 16/25$

$cos \theta = 4/5$

2

$\tan \theta = dfrac{sin \theta}{cos \theta}$

Connecting all three ratios

Since $sin = \text{opp}/\text{hyp}$ and $cos = \text{adj}/\text{hyp}$:

$dfrac{sin}{cos} = dfrac{\text{opp}/\text{hyp}}{\text{adj}/\text{hyp}} = dfrac{\text{opp}}{\text{adj}} = \tan$

This identity is useful when you know sine and cosine but need tangent.

3

Complementary Angles

Angles that add to 90°

In a right-angled triangle, the two acute angles add to 90°. They are complementary.

If one angle is $\theta$, the other is $90° - \theta$.

Key relationships:

  • $sin \theta = cos(90° - \theta)$
  • $cos \theta = sin(90° - \theta)$

Example: $sin 30° = cos 60° = 0.5$

✓

Check Understanding

Try it yourself

If $sin \theta = 5/13$, find $cos \theta$ and $\tan \theta$.

Worked Example

Trigonometric Ratios

1

If $cos \theta = 12/13$, find $sin \theta$.

$sin^2 = 1 - (12/13)^2 = 1 - 144/169 = 25/169$

$sin = 5/13$

2

Show that $\tan 30° = sin 30° / cos 30°$.

$sin 30° = 1/2$, $cos 30° = sqrt{3}/2$

$dfrac{1/2}{sqrt{3}/2} = dfrac{1}{sqrt{3}} = \tan 30°$ ✓

3

Verify $sin 40° = cos 50°$.

$40° + 50° = 90°$, so they are complementary.

Using calculator: $sin 40° approx 0.643$ and $cos 50° approx 0.643$ ✓

Common Misconceptions

✕

$sin^2 \theta$ means $sin(\theta^2)$. No, it means $(sin \theta)^2$, the square of the sine value.

✕

$sin 60° = 2 \times sin 30°$. No, trig ratios do not scale linearly. $sin 60° = sqrt{3}/2 approx 0.866$, while $2 \times 0.5 = 1$.

✕

$cos \theta$ can be greater than 1. No, since adjacent is always shorter than or equal to hypotenuse, $cos \theta$ is between 0 and 1.

Your Turn

Practice, Trig Ratios

Work through each question in your book or digitally. Answers are in the Questions phase.

1If $sin \theta = 4/5$, find $cos \theta$ and $\tan \theta$.
2Show that $sin^2 45° + cos^2 45° = 1$.
3Find $\theta$ if $sin \theta = cos \theta$.
Real-World Anchor

Astronomy and Navigation

Ancient astronomers used trigonometry to calculate distances to the moon and sun. Modern GPS still relies on trigonometric calculations to determine position from satellite signals. Australian surveyors use these principles for land measurement.

📓 Copy Into Your Books

▼

Identity

  • $sin^2 + cos^2 = 1$
  • $\tan = sin/cos$

Complementary

  • $sin \theta = cos(90-\theta)$
  • $cos \theta = sin(90-\theta)$

Given one ratio

  • Use identity to find second
  • Use $\tan = sin/cos$ for third
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
If $sin \theta = 3/5$, then $cos \theta = $
$sin^2 30° + cos^2 30° = $
$sin 40° = $
$\tan \theta = $
If $cos \theta = 0.8$, then $sin \theta = $
$sin 60° = cos$:
If $\tan \theta = 1$, then $\theta = $
$sin^2 \theta$ means:
1If $sin \theta = 5/13$, find $cos \theta$ and $\tan \theta$.
2Prove that $\tan \theta = sin \theta / cos \theta$ using a diagram.
3Find $\theta$ if $sin \theta = cos \theta$ and $0° < \theta < 90°$.

Comprehensive Answers

1$sin \theta = 4/5$, find $cos$ and $\tan$.
$cos = 3/5$, $\tan = 4/3$.
2Show $sin^2 45° + cos^2 45° = 1$.
Both $sin 45°$ and $cos 45°$ equal $1/sqrt{2}$. $(1/2) + (1/2) = 1$.
3Find $\theta$ if $sin \theta = cos \theta$.
$\theta = 45°$ (since $sin 45° = cos 45°$).
MC 1$sin = 3/5$, find $cos$.
$cos = 4/5$. Answer: B
MC 2$sin^2 + cos^2 = $.
$1$. Answer: B
MC 3$sin 40° = $.
$cos 50°$. Answer: B
MC 4$\tan = $.
$sin/cos$. Answer: B
MC 5$cos = 0.8$, find $sin$.
$sin = 0.6$. Answer: C
MC 6$sin 60° = cos$.
$30°$. Answer: A
MC 7$\tan \theta = 1$.
$45°$. Answer: B
MC 8Meaning of $sin^2$.
$(sin \theta)^2$. Answer: B
SA 1$sin = 5/13$, find $cos$ and $\tan$.
$cos = 12/13$, $\tan = 5/12$.
SA 2Prove $\tan = sin/cos$.
$sin/cos = (\text{opp}/\text{hyp})/(\text{adj}/\text{hyp}) = \text{opp}/\text{adj} = \tan$.
SA 3$sin \theta = cos \theta$.
$\theta = 45°$.
Game Phase
🎲
Game Unlocked!
You have mastered the Check Your Understanding questions. Choose a game mode below.
📦
Classify & Sort
Sort mathematical objects by their properties.
⚡
Speed Challenge
Answer questions against the clock.
📈
Match Maker
Match problems to their solutions.
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