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Math  /  Trigonometry

QuestionFind (a) sin⁡θ\sin \theta, (b) cos⁡θ\cos \theta, and (c) tan⁡θ\tan \theta for the given quadrantal angle. 11π11 \pi (a) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. sin⁡(11π)=\sin (11 \pi)= □\square (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The answer is undefined.

Studdy Solution

STEP 1

1. The angle 11π 11\pi is a quadrantal angle, meaning it lies on one of the axes of the unit circle.
2. We will use the properties of the unit circle to find the trigonometric values.

STEP 2

1. Determine the equivalent angle within the standard interval [0,2π)[0, 2\pi).
2. Find sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta using the unit circle.

STEP 3

First, we need to find the equivalent angle of 11π 11\pi within the interval [0,2π)[0, 2\pi). Since 11π 11\pi is greater than 2π 2\pi , we will reduce it by subtracting multiples of 2π 2\pi :
11π÷2π=5.5 11\pi \div 2\pi = 5.5
This means 11π 11\pi is 5×2π+1π 5 \times 2\pi + 1\pi . Therefore, the equivalent angle is:
11π−5×2π=π 11\pi - 5 \times 2\pi = \pi

STEP 4

Now that we have the equivalent angle π\pi, we will find the trigonometric values using the unit circle.
(a) sin⁡θ\sin \theta: The sine of π\pi is:
sin⁡(π)=0 \sin(\pi) = 0
(b) cos⁡θ\cos \theta: The cosine of π\pi is:
cos⁡(π)=−1 \cos(\pi) = -1
(c) tan⁡θ\tan \theta: The tangent of π\pi is:
tan⁡(π)=sin⁡(π)cos⁡(π)=0−1=0 \tan(\pi) = \frac{\sin(\pi)}{\cos(\pi)} = \frac{0}{-1} = 0
The values are: (a) sin⁡(11π)=0\sin(11\pi) = 0 (b) cos⁡(11π)=−1\cos(11\pi) = -1 (c) tan⁡(11π)=0\tan(11\pi) = 0

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