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Chapter 6: Problem 70

Find the reference angle and the exact function value if they exist. $$\sin \left(-135^{\circ}\right)$$

Short Answer

Expert verified
The reference angle for \(-135^{\circ}\) is \[ 45^{\circ} \. The exact value is \(- \frac{ \sqrt{2} }{ 2 }\).

Step by step solution

01

Understand the problem

The task is to find the reference angle for \(\begin{bmatrix} -135^{\circ} \end{bmatrix}\) and then compute the exact value of \[ \sin \left( -135^{\circ} \right) \].
02

Locate the angle on the unit circle

Starting from \[ 0^{\circ} \], move in the clockwise direction (since the angle is negative) by \(-135^{\circ}\). This places the angle in the third quadrant.
03

Find the reference angle

The reference angle is the positive acute angle formed by the terminal side of \(-135^{\circ}\) and the x-axis. Since \(-135^{\circ}\) is 45 degrees away from \[ -180^{\circ} \], the reference angle is \[ 45^{\circ} \].
04

Determine the sine value in the third quadrant

The sine of an angle in the third quadrant is negative. Using the reference angle \([45^{\circ}]\), where \ \sin \(45^{\circ} = \frac{ \sqrt{2} }{ 2 }\), the sine of \(-135^{\circ}\) is \(- \frac { \sqrt{2} } { 2 }\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

reference angle
To understand reference angles, first know that they help simplify problems in trigonometry. A reference angle is always a positive acute angle. That means it's between 0掳 and 90掳. It's formed between the terminal side of the given angle and the x-axis.
To find the reference angle for a negative angle like -135掳, first locate it on the unit circle. Move clockwise by 135掳. You'll see the terminal side ends in the third quadrant.
To get the reference angle, find the difference between the nearest x-axis and the given angle. For -135掳, subtract it from -180掳, resulting in a reference angle of 45掳.
unit circle
The unit circle is a circle with a radius of 1 unit centered at the origin (0,0) in the coordinate plane.
It's a key tool for understanding trigonometric functions. The angle in degrees or radians is measured from the positive x-axis.
When plotting negative angles, move clockwise. For positive angles, move counterclockwise.
For -135掳, starting from 0掳, move clockwise, which lands you in the third quadrant. Remember, every point (x, y) on the unit circle corresponds to cos(胃) and sin(胃) for that angle 胃.
sine function
The sine function defines the y-coordinate on the unit circle for any angle 胃. For example, sin(胃) equals the y-value at angle 胃.
In our case, sin(-135掳) can be found using its reference angle, 45掳, since they share the same sine value but may differ in sign depending on the quadrant.
At 45掳, sin(45掳)=鈭�2/2. Since -135掳 is in the third quadrant, sine is negative here. Thus, sin(-135掳)= -鈭�2/2.
quadrants
The coordinate plane is divided into four quadrants. Each quadrant affects the sign of trigonometric functions:
  • First Quadrant: All trigonometric functions are positive.
  • Second Quadrant: Sine is positive; cosine and tangent are negative.
  • Third Quadrant: Tangent is positive; sine and cosine are negative.
  • Fourth Quadrant: Cosine is positive; sine and tangent are negative.
Since -135掳 is in the third quadrant, we know sin(-135掳) is negative. Thus, it produces -鈭�2/2 for the sine function based on its reference angle.

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Most popular questions from this chapter

Given that \(\sec \beta=1.5304,\) find \(\sin \left(90^{\circ}-\beta\right)\)

Find the reference angle and the exact function value if they exist. $$\cos 90^{\circ}$$

Find the function value. Round to four decimal places. $$\cos 205.5^{\circ}$$

The transformation techniques that we learned in this section for graphing the sine and cosine functions can also be applied to the other trigonometric functions. Sketch a graph of each of the following. Then check your work using a graphing calculator. $$y=2 \csc \left(\frac{1}{2} x-\frac{3 \pi}{4}\right)$$

Find the reference angle and the exact function value if they exist. $$\cos 270^{\circ}$$
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