Trigonometry II

Introduction

In Trigonometry II, we delve deeper into the study of trigonometric functions and their applications. This topic builds on the foundational knowledge of trigonometry learned in previous classes and introduces more advanced concepts that are crucial for solving complex problems involving angles and triangles.

Basic Trigonometric Ratios

Trigonometric ratios are defined based on the relationships between the sides of a right-angled triangle. The three primary trigonometric ratios are:

• Sine ($\sin\theta$): opposite/hypotenuse
• Cosine ($\cos\theta$): adjacent/hypotenuse
• Tangent ($\tan\theta$): opposite/adjacent

Example: Given a right triangle with an angle $\theta$, where the opposite side is 5 and the hypotenuse is 13, find the sine of $\theta$. Solution: $$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13}$$

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all possible values of the variables. Some common trigonometric identities include:

• Pythagorean Identity: $\sin^2\theta + \cos^2\theta = 1$
• Even-Odd Identities: $\sin(-\theta) = -\sin\theta$, $\cos(-\theta) = \cos\theta$

Example: Prove the Pythagorean Identity $\sin^2\theta + \cos^2\theta = 1$. Solution: Starting with $\sin^2\theta + \cos^2\theta$, we substitute $\sin^2\theta = 1 - \cos^2\theta$, giving $1 - \cos^2\theta + \cos^2\theta = 1$ which simplifies to $1 = 1$, proving the identity.

Trigonometric Functions of Special Angles

Special angles like 30°, 45°, and 60° have exact trigonometric values that are commonly used in calculations. For example:

• $\sin 30° = \frac{1}{2}$
• $\cos 45° = \frac{\sqrt{2}}{2}$
• $\tan 60° = \sqrt{3}$

Example: Find the exact value of $\sin 135°$. Solution: Since 135° is in the second quadrant, the sine function is positive. By using the angle difference formula, we get $\sin 135° = \sin(90° + 45°) = \cos 45° = \frac{\sqrt{2}}{2}$.

Trigonometric Equations

Trigonometric equations involve trigonometric functions and are solved by applying trigonometric identities and properties. These equations often have multiple solutions within a specified domain.

Example: Solve the equation $\cos\theta = \sin\theta$ for $0° \leq \theta \leq 360°$. Solution: Using the Pythagorean Identity, we rewrite the equation as $\cos\theta = \sqrt{1 - \cos^2\theta}$. Squaring both sides gives $1 - \cos^2\theta = 1 - \sin^2\theta$, leading to $\sin^2\theta - \cos^2\theta = 0$. Factoring gives $(\sin\theta + \cos\theta)(\sin\theta - \cos\theta) = 0$, so $\theta = 45°$ or $\theta = 135°$.

Common Mistakes

• Forgetting to convert angles to the correct quadrant when finding trigonometric values.
• Misapplying trigonometric identities and formulas.
• Confusing between the different trigonometric ratios and their definitions.

Key Points

• Trigonometric ratios relate the sides of a right triangle.
• Trigonometric identities are essential for simplifying trigonometric expressions.
• Special angles have exact trigonometric values that are commonly used.
• Trigonometric equations involve solving for unknown angles using trigonometric functions.

Practice Questions

1. Find the value of $\tan 60°$.

Solution: $$\tan 60° = \frac{\sin 60°}{\cos 60°} = \frac{\sqrt{3}/2}{1/2} = \sqrt{3}$$

2. Prove the identity $\tan\theta = \frac{\sin\theta}{\cos\theta}$.

Solution: Starting with $\tan\theta = \frac{\sin\theta}{\cos\theta}$, we know that $\tan\theta = \frac{\sin\theta}{\cos\theta}$ by definition.

3. Solve the equation $\sin\theta = \cos\theta$ for $0° \leq \theta \leq 180°$.

Solution: Since $\sin\theta = \cos\theta$ is true for $\theta = 45°$, the solution is $\theta = 45°$.

4. Calculate the exact value of $\cos 120°$.

Solution: Since 120° is in the second quadrant, $\cos 120° = -\cos(180° - 120°) = -\cos 60° = -\frac{1}{2}$.

5. Find the value of $\sin 150°$.

Solution: Since 150° is in the second quadrant, $\sin 150° = \sin(180° - 30°) = \sin 30° = \frac{1}{2}$.