Problem 23

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. $$ t=-5 \pi / 6 $$

Step-by-Step Solution

Verified

Answer

The reference angle is $ \frac{\pi}{6} $, $ \sin t = -\frac{1}{2} $, and $ \cos t = -\frac{\sqrt{3}}{2} $.

1Step 1: Determine the reference angle

To find the reference angle for the angle $ t = -\frac{5\pi}{6} $, sketch the angle in standard position starting from the positive x-axis and rotating clockwise. The angle $-\frac{5\pi}{6}$ is equivalent to rotating by $\frac{5\pi}{6}$ counterclockwise in the opposite quadrant. Hence, the reference angle $ t' $ is simply $ \frac{\pi}{6} $ because it is the smallest angle with the x-axis.

2Step 2: Determine the sign of sine and cosine

Since $-\frac{5\pi}{6}$ is in the third quadrant where both sine and cosine are negative, the sine and cosine values of $ t $ will have their reference angle values but will be negative.

3Step 3: Calculate $ \sin t $ using reference angle

The sine of $ \frac{\pi}{6} $ is $ \frac{1}{2} $. Therefore, the sine of $ t = -\frac{5\pi}{6} $ is $ \sin(-\frac{5\pi}{6}) = -\frac{1}{2} $ because it is negative in the third quadrant.

4Step 4: Calculate $ \cos t $ using reference angle

The cosine of $ \frac{\pi}{6} $ is $ \frac{\sqrt{3}}{2} $. Therefore, the cosine of $ t = -\frac{5\pi}{6} $ is $ \cos(-\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2} $ because it is also negative in the third quadrant.

Key Concepts

Reference AngleSine FunctionCosine Function

Reference Angle

A reference angle is an important concept in trigonometry used to simplify calculations involving angles. When you're given any angle, especially one larger than 90 degrees, finding a reference angle allows you to determine trigonometric functions based on known values from the first quadrant. The reference angle itself is the smallest positive angle between the terminal side of the angle and the x-axis.

To find the reference angle of an angle such as $t = -\frac{5\pi}{6}$, imagine turning clockwise from the positive x-axis, as the negative sign indicates a clockwise direction.

The equivalent counterclockwise angle, $\frac{5\pi}{6}$, lies in the third quadrant when referred to a standard position.

Thus, the reference angle is calculated as follows: Because $\frac{5\pi}{6}$ would rest in the second quadrant, its reference angle remains $\frac{\pi}{6}$.

Once found, the reference angle makes computing trigonometric functions more straightforward since the values in the first quadrant are commonly memorized or easily accessible.

Sine Function

The sine function is a fundamental trigonometric function that relates an angle of a right triangle to the ratio of the lengths of the opposite side to the hypotenuse. For the unit circle, it describes the y-coordinate of a point as you move counterclockwise around the circle.

To determine $\sin(t)$ for an angle like $t = -\frac{5\pi}{6}$, you use the reference angle $\frac{\pi}{6}$.

The sine of $\frac{\pi}{6}$ is known to be $\frac{1}{2}$.

In the third quadrant where the angle $ t = -\frac{5\pi}{6} $ lies, the sine function is negative, so $\sin(-\frac{5\pi}{6}) = -\frac{1}{2}$.

Through understanding in which quadrant an angle lies, students can better predict the sign of sine values, enhancing their proficiency with trigonometric problems.

Cosine Function

The cosine function, another key trigonometric function, involves the ratio of the adjacent side of a triangle to its hypotenuse. On the unit circle, this equates to the x-coordinate as you move around the circle.