Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the key areas of study in trigonometry is the exact values of sine, cosine, and tangent for specific angles. This knowledge is essential for solving various mathematical problems, especially in geometry and physics.

Introduction

In trigonometry, the exact values of sine, cosine, and tangent for specific angles are crucial for solving problems efficiently. This section focuses on memorizing these values for angles commonly encountered in mathematical contexts.

Sine, Cosine, and Tangent

These fundamental trigonometric functions relate angles of a triangle to its side lengths. For a right-angled triangle:

• Sine (sin): ratio of opposite side to hypotenuse.
• Cosine (cos): ratio of adjacent side to hypotenuse.
• Tangent (tan): ratio of opposite side to adjacent side.

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Exact Values for Specific Angles

Memorize these values as they frequently arise in various mathematical problems.

0° and 90°

• $\text{ sin 0°} = 0, \text{ cos 0° } = 1, \text{ tan 0° } = 0$
• $\text{ sin 90° } = 1, \text{ cos 90° } = 0, \text{ tan 90° } \text{is} \text{ undefined}$

30°, 45°, and 60°

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

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Worked Examples

Example 1: Calculating the sine of 30°

Problem: Find the value of sin 30°.

Solution:

$\sin(30^\circ) = \frac{1}{2}$

Example 2: Using exact values in equations

Problem: Solve for x in $\sin(x) = \sqrt{3}/2$.

Solution:

$\sin(x) = \frac{\sqrt{3}}{2}$

Given this, we look for the angle whose sine value matches $\sqrt{3}/2$, which is 60°.

$x = 60^\circ$

Example 3: Finding the angle with a given cosine value

Problem: Find the angle whose cosine is $\frac{1}{2}.$

Solution:

$\cos(x) = \frac{1}{2}$

From the memorized values, $\cos(60^\circ) = \frac{1}{2}.$

$x = 60^\circ$

Practice Problems

Problem 1: Tangent of 45°

Given $\tan(45^\circ) = 1$, calculate the value.

Answer: $\tan(45^\circ) = 1$

Problem 2: Cosine of 30°

Find $\cos(30^\circ)).$

Answer: $\cos(30^\circ) = \dfrac{\sqrt{3}}{2}$

Problem 3: Angle with sine of 0

Identify the angle whose sine value is $0.$

Answer: $\sin(x) = 0 \Rightarrow x = 0^\circ \text{ or } 180^\circ$