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Math

Question Find the missing trigonometric function values for a 30° angle without using a calculator.

Studdy Solution

STEP 1

Assumptions
1. We are working with a standard unit circle and the commonly known exact values for trigonometric functions at specific angles.
2. The angle provided is 30∘30^{\circ}.
3. The trigonometric functions we need to find are sin⁡θ\sin \theta, cot⁡θ\cot \theta, and sec⁡θ\sec \theta.
4. We will use the Pythagorean identities and the reciprocal identities of trigonometric functions.

STEP 2

We will start by finding the value of sin⁡θ\sin \theta using the known value of cos⁡θ\cos \theta and the Pythagorean identity:
sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

STEP 3

Substitute the known value of cos⁡θ=32\cos \theta = \frac{\sqrt{3}}{2} into the identity:
sin⁡2θ+(32)2=1\sin^2 \theta + \left(\frac{\sqrt{3}}{2}\right)^2 = 1

STEP 4

Simplify the square of cos⁡θ\cos \theta:
sin⁡2θ+34=1\sin^2 \theta + \frac{3}{4} = 1

STEP 5

Isolate sin⁡2θ\sin^2 \theta:
sin⁡2θ=1−34\sin^2 \theta = 1 - \frac{3}{4}

STEP 6

Subtract to find sin⁡2θ\sin^2 \theta:
sin⁡2θ=14\sin^2 \theta = \frac{1}{4}

STEP 7

Take the square root of both sides to find sin⁡θ\sin \theta. Since sin⁡30∘\sin 30^{\circ} is positive in the first quadrant, we only consider the positive root:
sin⁡θ=12\sin \theta = \frac{1}{2}

STEP 8

Next, we will find the value of cot⁡θ\cot \theta using the known value of tan⁡θ\tan \theta and the reciprocal identity:
cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}

STEP 9

Substitute the known value of tan⁡θ=33\tan \theta = \frac{\sqrt{3}}{3} into the identity:
cot⁡θ=133\cot \theta = \frac{1}{\frac{\sqrt{3}}{3}}

STEP 10

Simplify the expression to find cot⁡θ\cot \theta:
cot⁡θ=33\cot \theta = \frac{3}{\sqrt{3}}

STEP 11

Rationalize the denominator:
cot⁡θ=33⋅33\cot \theta = \frac{3}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}

STEP 12

Simplify the expression:
cot⁡θ=3\cot \theta = \sqrt{3}

STEP 13

Finally, we will find the value of sec⁡θ\sec \theta using the known value of cos⁡θ\cos \theta and the reciprocal identity:
sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

STEP 14

Substitute the known value of cos⁡θ=32\cos \theta = \frac{\sqrt{3}}{2} into the identity:
sec⁡θ=132\sec \theta = \frac{1}{\frac{\sqrt{3}}{2}}

STEP 15

Simplify the expression to find sec⁡θ\sec \theta:
sec⁡θ=23\sec \theta = \frac{2}{\sqrt{3}}

STEP 16

Rationalize the denominator:
sec⁡θ=23⋅33\sec \theta = \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}

STEP 17

Simplify the expression:
sec⁡θ=233\sec \theta = \frac{2\sqrt{3}}{3}

STEP 18

Now we can complete the table with the exact trigonometric function values we have found:
\begin{tabular}{|c|c|c|c|c|c|c|} \hlineθ\theta & sin⁡θ\sin \theta & cos⁡θ\cos \theta & tan⁡θ\tan \theta & cot⁡θ\cot \theta & sec⁡θ\sec \theta & csc⁡θ\csc \theta \\ \hline 30∘30^{\circ} & 12\frac{1}{2} & 32\frac{\sqrt{3}}{2} & 33\frac{\sqrt{3}}{3} & 3\sqrt{3} & 233\frac{2\sqrt{3}}{3} & 2 \\ \hline \end{tabular}

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