Problem 22 - Precalculus with Calculus Previews

Question

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. $$ t=-7 \pi / 4 $$

Step-by-Step Solution

Answer

The reference angle is \( \frac{\pi}{4} \), \( \sin t = \frac{\sqrt{2}}{2} \), and \( \cos t = \frac{\sqrt{2}}{2} \).

Step 1: Determine Equivalent Positive Angle

Given the angle \( t = -\frac{7\pi}{4} \), we need to find an equivalent positive angle by adding \( 2\pi \) until we obtain a positive angle. Adding \( 2\pi \), which is \( \frac{8\pi}{4} \), gives us \( t = -\frac{7\pi}{4} + \frac{8\pi}{4} = \frac{\pi}{4} \).

Step 2: Determine Reference Angle

The reference angle \( t^{\prime} \) of an angle \( t \) is the acute angle formed by the terminal side of \( t \) and the horizontal axis. Since \( \frac{\pi}{4} \) is already between 0 and \( \pi \), \( t^{\prime} = \frac{\pi}{4} \).

Step 3: Calculate Exact Value of Sine

The sine of an angle \( \frac{\pi}{4} \), which is in the first quadrant, is positive. We use the reference angle to find \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Step 4: Calculate Exact Value of Cosine

Similarly, the cosine of an angle \( \frac{\pi}{4} \) is also positive in the first quadrant. Calculate \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Key Concepts

Understanding Reference Angles

A reference angle is the smallest angle that the terminal side of your given angle makes with the x-axis. Picture this: if you have an angle, and it doesn't land in the first quadrant (0 to \( \pi/2 \)), you can find its reference angle, which will always be between 0 and \( \pi/2 \). This makes it easier to work out trigonometric function values using basic triangle properties.

To determine a reference angle, first ensure that your angle falls between 0 and \( 2\pi \) through equivalent positive angles if needed. Once in this range, if it falls in the first quadrant, the angle is already the reference angle. For angles that fall in quadrants two, three, or four, subtract the angle from \( \pi \) (second quadrant), or subtract it from \( \pi \) and then from \( 2\pi \) for third and fourth quadrants respectively.

This approach simplifies calculations regarding the sine and cosine of angles located in different quadrants.

Exact Sine and Cosine Values

The sine and cosine values of angles are calculated using their reference angles, especially when dealing with common angles like \( \pi/4 \), \( \pi/3 \), \( \pi/6 \), etc.

If the reference angle is \( \pi/4 \), then both \( \sin(\pi/4) \) and \( \cos(\pi/4) \) have exact values of \( \frac{\sqrt{2}}{2} \).

Let's highlight some key reasons why these values are useful:

• They are exact and eliminate the need for decimal approximations.
• The values are derived from special right triangles, which are foundational in trigonometry.

In solving problems involving trigonometric functions, knowing the reference angle allows predictions about the function's sign based on the quadrant location of the original angle.

For angles in the first quadrant, like \( \pi/4 \), both sine and cosine values are positive.

Creating Equivalent Angles

To work efficiently with trigonometric functions, sometimes you will have to convert your given angle to an equivalent angle that can be more easily analyzed. An equivalent angle is simply the angle measured from the positive x-axis, adjusting by multiples of \( 2\pi \) if necessary.

For angles given in degrees, you add or subtract 360 degrees to end up with a positive angle. In our case with angles in radians, you add or subtract \( 2\pi \) until the angle is within the range of 0 to \( 2\pi \).

Equivalent angles are crucial because they translate complex angles into simpler forms without altering their trigonometric implications. For example, \(-\frac{7\pi}{4}\) becomes \(\frac{\pi}{4}\) after adding \( 2\pi \), landing it nicely in the first quadrant where standard triangle rules apply.

• This adjustment reflects the periodic nature of trigonometric functions.
• Recognizing equivalent angles simplifies evaluating sine and cosine functions.

Understanding equivalent angles ensures that you're always working with the most straightforward version of any given angle.