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Chapter 4: Problem 18

For the given value of \(t\) determine the reference angle \(t^{\prime}\) and the exact values of \(\sin t\) and \(\cos t\). Do not use a calculator. $$ t=3 \pi / 4 $$

Short Answer

Expert verified
Reference angle is \( \frac{\pi}{4} \); \( \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \), \( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \).

Step by step solution

01

Identify the Reference Angle

The reference angle \( t' \) for an angle \( t \) in radians is found by calculating the distance from \( t \) to the nearest x-axis in the unit circle. For an angle in the second quadrant, the reference angle \( t' = \pi - t \). In this case, \( t = \frac{3\pi}{4} \), so \( t' = \pi - \frac{3\pi}{4} = \frac{\pi}{4} \).
02

Find the Sine of the Angle

The sine of an angle in the unit circle is positive in both the first and second quadrants. Because \( t = \frac{3\pi}{4} \) is in the second quadrant, \( \sin t = \sin(\frac{\pi}{4}) \). The sine of \( \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \), so \( \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \).
03

Find the Cosine of the Angle

The cosine of an angle in the unit circle is negative in the second quadrant. Thus, \( \cos t = -\cos(\frac{\pi}{4}) \). The cosine of \( \frac{\pi}{4} \) is also \( \frac{\sqrt{2}}{2} \), but due to the quadrant, \( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \).

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Key Concepts

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

Reference Angle
In trigonometry, a reference angle is a positive acute angle that helps us find the coordinates of a given non-acute angle in standard position on the unit circle. It is closely related to the concept of symmetry in the unit circle.
  • The reference angle (\( t' \)) should be between \( 0 \) and \( \frac{\pi}{2} \) radians, which corresponds to 0 and 90 degrees.
  • To find the reference angle in different quadrants, use different rules based on the location of the angle:
    • First Quadrant: \( t' = t \)
    • Second Quadrant: \( t' = \pi - t \)
    • Third Quadrant: \( t' = t - \pi \)
    • Fourth Quadrant: \( t' = 2\pi - t \)
  • The reference angle is useful for determining the sine, cosine, and other trigonometric function values of any angle using its reference in the first quadrant.
Unit Circle
The unit circle is a foundational concept in trigonometry that represents the set of all points \((x, y)\) satisfying \( x^2 + y^2 = 1 \). Visualizing the unit circle can help you understand trigonometric functions and angles.
  • Each point on the unit circle corresponds to an angle \( t \) in radians, measured counterclockwise from the positive x-axis.
  • The unit circle has a radius of 1, making calculation of trigonometric values straightforward.
  • The angle \( t \) also describes a point \((\cos t, \sin t)\) on the circle.
  • This makes the unit circle a great tool for memorizing trigonometric values at key angles, such as \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), \( \frac{\pi}{2} \), and others.
Sine
The sine function gives the y-coordinate of a point on the unit circle. It is especially important to identify in which quadrant the angle is located.
  • For angles in the first quadrant, \( \sin t \) is positive and equals the height of a right triangle formed with the x-axis.
  • Sine values remain positive in the second quadrant because the y-coordinate is still positive.
  • If the reference angle is \( \frac{\pi}{4} \), then \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).
  • Therefore, for an angle \( t = \frac{3\pi}{4} \) in the second quadrant, \( \sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2} \).
Cosine
The cosine function gives the x-coordinate on the unit circle and is key for understanding the horizontal displacement of an angle's terminal side.
  • For first quadrant angles, \( \cos t \) is positive as the x-coordinate is positive.
  • The x-coordinate shifts to negative in the second quadrant, making \( \cos t \) negative.
  • If the reference angle is \( \frac{\pi}{4} \), then \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).
  • Thus, for an angle \( t = \frac{3\pi}{4} \) located in the second quadrant, \( \cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} \).

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Most popular questions from this chapter

Use a product-to-sum formula in Theorem 4.7 .1 to write the given product as a sum of cosines or a sum of sines. $$ \sin \pi \theta \cos 7 \pi \theta $$

In Problems 27 and 28, use a calculator or computer to estimate the given limit by completing each table. Round the entries in each table to eight decimal places. $$ \lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}} $$

Use a product-to-sum formula in Theorem 4.7 .1 to write the given product as a sum of cosines or a sum of sines. $$ 2 \sin \left(x+\frac{\pi}{4}\right) \sin \left(x-\frac{\pi}{4}\right) $$

Describe in words how you would obtain the graph of the given function by starting with the graph of \(y=\sin x\) (Problem 65 ) and the graph of \(y=\cos x(\) Problem 66\()\). $$ y=-6+\frac{1}{4} \cos \left(\frac{1}{2} x+\pi\right) $$

Use a graphing utility to investigate whether the given function is periodic. $$ f(x)=\frac{1}{\sin 2 x} $$
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