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Math  /  Trigonometry

QuestionFind cot⁡θ,sec⁡θ\cot \theta, \sec \theta, and cos⁡θ\cos \theta, where θ\theta is the angle shown in the figure. Give exact values, not decimal approximations. sec⁡θ=cos⁡θ=\begin{array}{l} \sec \theta= \\ \cos \theta= \end{array}

Studdy Solution

STEP 1

1. The triangle is a right triangle.
2. The side adjacent to θ\theta is 4 units long.
3. The hypotenuse is 5 units long.
4. The Pythagorean theorem can be used to find the missing side.

STEP 2

1. Use the Pythagorean theorem to find the length of the opposite side.
2. Calculate cos⁡θ\cos \theta.
3. Calculate sec⁡θ\sec \theta.
4. Calculate cot⁡θ\cot \theta.

STEP 3

Apply the Pythagorean theorem: a2+b2=c2 a^2 + b^2 = c^2 , where a a and b b are the legs of the triangle, and c c is the hypotenuse. We have:
42+b2=52 4^2 + b^2 = 5^2

STEP 4

Calculate the square of the known sides:
16+b2=25 16 + b^2 = 25

STEP 5

Solve for b2 b^2 :
b2=25−16 b^2 = 25 - 16 b2=9 b^2 = 9

STEP 6

Take the square root to find b b :
b=9 b = \sqrt{9} b=3 b = 3

STEP 7

Calculate cos⁡θ\cos \theta using the definition cos⁡θ=adjacenthypotenuse\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}:
cos⁡θ=45 \cos \theta = \frac{4}{5}

STEP 8

Calculate sec⁡θ\sec \theta as the reciprocal of cos⁡θ\cos \theta:
sec⁡θ=1cos⁡θ=145=54 \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{4}{5}} = \frac{5}{4}

STEP 9

Calculate cot⁡θ\cot \theta using the definition cot⁡θ=adjacentopposite\cot \theta = \frac{\text{adjacent}}{\text{opposite}}:
cot⁡θ=43 \cot \theta = \frac{4}{3}
The exact values are: sec⁡θ=54\sec \theta = \frac{5}{4} cos⁡θ=45\cos \theta = \frac{4}{5} cot⁡θ=43\cot \theta = \frac{4}{3}

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