Math

QuestionFind the values of the trigonometric functions given that sinθ=1\sin \theta=1. What is cosθ\cos \theta? A. cosθ=\cos \theta= B. Undefined.

Studdy Solution

STEP 1

Assumptions1. The sine of theta is given as1. . We are looking for the exact values of the remaining trigonometric functions of theta.
3. We are working within the unit circle, where the radius is1.
4. The sine of theta represents the y-coordinate of the point where the terminal side of the angle intersects the unit circle.
5. The cosine of theta represents the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
6. The tangent of theta is the ratio of the sine to the cosine.
7. The secant, cosecant, and cotangent are the reciprocals of the cosine, sine, and tangent, respectively.

STEP 2

Since sinθ=1\sin \theta =1, this means that the angle θ\theta is at the point where the terminal side intersects the unit circle at the y-coordinate of1. This occurs at θ=π2\theta = \frac{\pi}{2} or θ=90\theta =90^{\circ}.

STEP 3

Let's find the cosine of theta. The cosine of theta represents the x-coordinate of the point where the terminal side of the angle intersects the unit circle. At θ=π2\theta = \frac{\pi}{2} or θ=90\theta =90^{\circ}, the x-coordinate is0. So, cosθ=0\cos \theta =0.

STEP 4

Next, let's find the tangent of theta. The tangent of theta is the ratio of the sine to the cosine. So, tanθ=sinθcosθ\tan \theta = \frac{\sin \theta}{\cos \theta}.

STEP 5

Plug in the values for the sine and cosine of theta to calculate the tangent of theta.
tanθ=10\tan \theta = \frac{1}{0}

STEP 6

The expression 10\frac{1}{0} is undefined, so the tangent of theta is undefined.

STEP 7

Next, let's find the secant of theta. The secant of theta is the reciprocal of the cosine. So, secθ=1cosθ\sec \theta = \frac{1}{\cos \theta}.

STEP 8

Plug in the value for the cosine of theta to calculate the secant of theta.
secθ=10\sec \theta = \frac{1}{0}

STEP 9

The expression \frac{}{} is undefined, so the secant of theta is undefined.

STEP 10

Next, let's find the cosecant of theta. The cosecant of theta is the reciprocal of the sine. So, cscθ=sinθ\csc \theta = \frac{}{\sin \theta}.

STEP 11

Plug in the value for the sine of theta to calculate the cosecant of theta.
cscθ=\csc \theta = \frac{}{}

STEP 12

Calculate the cosecant of theta.
cscθ=\csc \theta =

STEP 13

Finally, let's find the cotangent of theta. The cotangent of theta is the reciprocal of the tangent. So, cotθ=tanθ\cot \theta = \frac{}{\tan \theta}.

STEP 14

Since the tangent of theta is undefined, the cotangent of theta is also undefined.
So, the exact values of the remaining trigonometric functions of θ\theta areA. cosθ=0\cos \theta =0 B. tanθ\tan \theta is undefinedC. secθ\sec \theta is undefined. cscθ=\csc \theta = . cotθ\cot \theta is undefined

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