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Math  /  Trigonometry

QuestionEvaluate (if possible) the sine, cosine, and tangent at the real number tt. (If an answer is undetined, t=π3t=\frac{\pi}{3} sin⁡t=\sin t= □\square cos⁡t=\cos t= □\square tan⁡t=\tan t= □\square

Studdy Solution

STEP 1

1. We are given the angle t=π3 t = \frac{\pi}{3} .
2. We need to evaluate the sine, cosine, and tangent of t t .
3. We will use known values from the unit circle for standard angles.

STEP 2

1. Evaluate sin⁡t\sin t.
2. Evaluate cos⁡t\cos t.
3. Evaluate tan⁡t\tan t.

STEP 3

Recall the sine value for the angle π3 \frac{\pi}{3} .
The sine of π3 \frac{\pi}{3} is a known value from the unit circle:
sin⁡(π3)=32 \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}

STEP 4

Recall the cosine value for the angle π3 \frac{\pi}{3} .
The cosine of π3 \frac{\pi}{3} is a known value from the unit circle:
cos⁡(π3)=12 \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}

STEP 5

Evaluate the tangent of t t using the sine and cosine values.
The tangent is defined as the ratio of sine to cosine:
tan⁡(π3)=sin⁡(π3)cos⁡(π3) \tan\left(\frac{\pi}{3}\right) = \frac{\sin\left(\frac{\pi}{3}\right)}{\cos\left(\frac{\pi}{3}\right)}
Substitute the known values:
tan⁡(π3)=3212 \tan\left(\frac{\pi}{3}\right) = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}
Simplify the expression:
tan⁡(π3)=3 \tan\left(\frac{\pi}{3}\right) = \sqrt{3}
The values are:
sin⁡t=32\sin t = \frac{\sqrt{3}}{2} cos⁡t=12\cos t = \frac{1}{2} tan⁡t=3\tan t = \sqrt{3}

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