Question

Use a coterminal angle to find the exact value of each expression. Do not use a calculator. $$ \sin 405^{\circ} $$

Short answer

\( \sin 405^{°} = \frac{\

Step-by-step solution

Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. To find a coterminal angle, you can add or subtract multiples of 360° from the given angle.

Find a Coterminal Angle of 405°

To find a coterminal angle for 405°, subtract 360° until the angle is between 0° and 360°.\( 405^{\text{°}} − 360^{\text{°}} = 45^{\text{°}} \)Thus, 45° is a coterminal angle of 405°.

Use the Known Value of Sine for 45°

Use the exact value of the sine function for the coterminal angle found above. \( \sin 45^{\text{°}} = \frac{ \sqrt{2} }{2} \)

Key concepts

Sine Function

The sine function, denoted as \(\text{sin}\), is one of the primary trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. The general formula is: \(\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
The sine function is periodic with a period of 360° (or 2\(\text{π}\) radians). This means that every 360°, the sine function repeats its values.
Knowing the periodic nature of the sine function is crucial when working with coterminal angles. It ensures that the sine value of \(\theta\) will be the same as the sine value of \(\theta + 360^{\text{°}}\) or \(\theta - 360^{\text{°}}\).

Angle Measurement

Angles are measured in degrees or radians. A full circle is 360° or 2\(\text{π}\) radians.
Understanding how to measure angles and convert them is essential in trigonometry. For instance, the angle 405° exceeds one full rotation (360°), and so, by subtracting 360° from 405°, you can find a coterminal angle: 405° - 360° = 45°.
This coterminal angle is within the standard range (0° to 360°), making it easier to evaluate the sine function.
It's important to be familiar with common angle measures like 30°, 45°, 60°, and 90°, since trig values for these angles are frequently used.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved. They are essential tools for simplifying and solving trigonometric expressions.
Some key trigonometric identities include:

• Pythagorean Identity: \(\text{sin}^2(\theta) + \text{cos}^2(\theta) = 1\)
• Co-function Identity: \(\text{sin}(90^{\text{°}} - \theta) = \text{cos}(\theta)\)
• Angle Sum and Difference Identities: \(\text{sin}(\theta \pm \phi) = \text{sin}(\theta)\text{cos}(\text{φ}) \pm \text{cos}(\theta)\text{sin}(\text{φ})\)

In the given exercise, knowing that \(\text{sin}(45^{\text{°}}) = \frac{ \sqrt{2} }{2}\) is an application of the basic angle values and their sine functions, making it easier to solve problems without a calculator.