Question

Use the even-odd properties to find the exact value of each expression. Do not use a calculator. $$ \sin \left(-135^{\circ}\right) $$

Short answer

\( \sin(-135^{\circ}) = -\frac{\sqrt{2}}{2} \)

Step-by-step solution

Understand the Even-Odd Properties

The sine function is an odd function, which means \( \sin(-x) = -\sin(x) \). This property will help simplify the calculation.

Apply the Even-Odd Property

Use the odd property of the sine function to rewrite \( \sin(-135^{\circ}) \) as \( -\sin(135^{\circ}) \).

Find \( \sin(135^{\circ}) \)

135 degrees is in the second quadrant where sine is positive. It can be expressed as \( 180^{\circ} - 45^{\circ} \). Knowing that \( \sin(180^{\circ} - x) = \sin(x) \), we have \( \sin(135^{\circ}) = \sin(45^{\circ}) \).

Evaluate \( \sin(45^{\circ}) \)

\( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \).

Combine Results

Combine the results from the previous steps to get \( \sin(-135^{\circ}) = -\sin(135^{\circ}) = -\frac{\sqrt{2}}{2} \).

Key concepts

Sine Function

The sine function, denoted as \( \sin(x) \), is one of the fundamental trigonometric functions. It relates an angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. The sine function is periodic with a period of \( 360^{\circ} \) or \( 2\pi \) radians and is defined for all real numbers.
Key properties include:

• Range: \( -1 \leq \sin(x) \leq 1 \)
• Periodicity: \( \sin(x + 360^{\circ}) = \sin(x) \)
• Symmetry: Sine exhibits both odd and even properties in different contexts

Understanding these properties helps in solving problems involving trigonometric functions.

Even-Odd Properties

In mathematics, functions can be classified based on their symmetry. An odd function satisfies the condition \( f(-x) = -f(x) \) for all \( x \). The sine function is an odd function since \( \sin(-x) = -\sin(x) \).
This property simplifies calculations significantly. For example, if you need to find \( \sin(-135^{\circ}) \), you can use the odd property to rewrite it as \( -\sin(135^{\circ}) \). Recognizing and applying this symmetry can make complex problems much easier to solve.

Quadrant Analysis

The coordinate plane is divided into four quadrants, and the sign of trigonometric functions depends on the quadrant in which the angle lies.

• First Quadrant: \( 0^{\circ} \) to \( 90^{\circ} \) where all trigonometric functions are positive
• Second Quadrant: \( 90^{\circ} \) to \( 180^{\circ} \) where sine is positive, and cosine and tangent are negative
• Third Quadrant: \( 180^{\circ} \) to \( 270^{\circ} \) where tangent is positive, and sine and cosine are negative
• Fourth Quadrant: \( 270^{\circ} \) to \( 360^{\circ} \) where cosine is positive, and sine and tangent are negative

In our problem, \( 135^{\circ} \) is in the Second Quadrant where sine is positive. This quadrant analysis helps determine the sign and value of trigonometric functions.

Reference Angles

A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. It is always a positive acute angle (less than \( 90^{\circ} \)).
For instance, to find the value of \( \sin(135^{\circ}) \), we can use its reference angle. Since \( 135^{\circ} = 180^{\circ} - 45^{\circ} \), the reference angle for \( 135^{\circ} \) is \( 45^{\circ} \).
Knowing that \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \), and using the property \( \sin(180^{\circ} - x) = \sin(x) \), we get \( \sin(135^{\circ}) = \sin(45^{\circ}) \).
Understanding reference angles helps in evaluating the trigonometric function of any angle regardless of its location.