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QuestionMatch each trigonometric function with its value:
sin 60° = ?, cos 45° = ?, tan 45° = ?, sec 30° = ?, csc 30° = ?, cot 60° = ?
Choices: 22,22,233,1,32,33,32\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, \frac{2 \sqrt{3}}{3}, 1, \frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{2}

Studdy Solution

STEP 1

Assumptions1. The trigonometric functions are defined as follows - sin⁡θ\sin \theta is the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. - cos⁡θ\cos \theta is the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle. - tan⁡θ\tan \theta is the ratio of the length of the side opposite the angle to the length of the adjacent side in a right triangle. - sec⁡θ\sec \theta is the reciprocal of cos⁡θ\cos \theta. - csc⁡θ\csc \theta is the reciprocal of sin⁡θ\sin \theta. - cot⁡θ\cot \theta is the reciprocal of tan⁡θ\tan \theta. . The angles are given in degrees.
3. The values of the trigonometric functions for the given angles can be found using a unit circle or a right triangle with the given angle.

STEP 2

Let's start with the first function, sin⁡60∘\sin60^{\circ}. The value of sin⁡60∘\sin60^{\circ} is 2\frac{\sqrt{}}{2}.

STEP 3

Next, we have cos⁡45∘\cos45^{\circ}. The value of cos⁡45∘\cos45^{\circ} is 22\frac{\sqrt{2}}{2}.

STEP 4

The next function is tan⁡45∘\tan45^{\circ}. The value of tan⁡45∘\tan45^{\circ} is 11.

STEP 5

Next, we have sec⁡30∘\sec30^{\circ}. The secant function is the reciprocal of the cosine function. So, sec⁡30∘\sec30^{\circ} is the reciprocal of cos⁡30∘\cos30^{\circ}. The value of cos⁡30∘\cos30^{\circ} is 32\frac{\sqrt{3}}{2}, so sec⁡30∘\sec30^{\circ} is 23\frac{2}{\sqrt{3}}, which simplifies to 233\frac{2 \sqrt{3}}{3}.

STEP 6

The next function is csc⁡30∘\csc30^{\circ}. The cosecant function is the reciprocal of the sine function. So, csc⁡30∘\csc30^{\circ} is the reciprocal of sin⁡30∘\sin30^{\circ}. The value of sin⁡30∘\sin30^{\circ} is 12\frac{1}{2}, so csc⁡30∘\csc30^{\circ} is 22.

STEP 7

The last function is cot⁡60∘\cot60^{\circ}. The cotangent function is the reciprocal of the tangent function. So, cot⁡60∘\cot60^{\circ} is the reciprocal of tan⁡60∘\tan60^{\circ}. The value of tan⁡60∘\tan60^{\circ} is 3\sqrt{3}, so cot⁡60∘\cot60^{\circ} is 13\frac{1}{\sqrt{3}}, which simplifies to 33\frac{\sqrt{3}}{3}.
So, the final matches aresin⁡60∘=32cos⁡45∘=22tan⁡45∘=1sec⁡30∘=233csc⁡30∘=2cot⁡60∘=33\begin{array}{l} \sin60^{\circ}= \frac{\sqrt{3}}{2} \\ \cos45^{\circ}= \frac{\sqrt{2}}{2} \\ \tan45^{\circ}=1 \\ \sec30^{\circ}= \frac{2 \sqrt{3}}{3} \\ \csc30^{\circ}=2 \\ \cot60^{\circ}= \frac{\sqrt{3}}{3} \end{array}

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