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QuestionFind the values of secant and cosecant for θ=60∘\theta=60^{\circ} using trig function reciprocals. sec⁡(60∘)=\sec(60^{\circ})= csc⁡(60∘)=\csc(60^{\circ})=

Studdy Solution

STEP 1

Assumptions1. We have a trigonometric table at our disposal. . The angle θ\theta is given as 60∘60^{\circ}.
3. We know that the secant function is the reciprocal of the cosine function, and the cosecant function is the reciprocal of the sine function.

STEP 2

First, we need to find the cosine and sine values for the angle 60∘60^{\circ} from the trigonometric table.
cos⁡(60∘)=12\cos(60^{\circ}) = \frac{1}{2}sin⁡(60∘)=2\sin(60^{\circ}) = \frac{\sqrt{}}{2}

STEP 3

Now, we can calculate the secant of θ\theta by taking the reciprocal of the cosine of θ\theta.
sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}

STEP 4

Substitute the value of cos⁡(60∘)\cos(60^{\circ}) into the equation to calculate sec⁡(θ)\sec(\theta).
sec⁡(60∘)=1cos⁡(60∘)=112\sec(60^{\circ}) = \frac{1}{\cos(60^{\circ})} = \frac{1}{\frac{1}{2}}

STEP 5

Calculate the value of sec⁡(60∘)\sec(60^{\circ}).
sec⁡(60∘)=112=2\sec(60^{\circ}) = \frac{1}{\frac{1}{2}} =2

STEP 6

Next, we can calculate the cosecant of θ\theta by taking the reciprocal of the sine of θ\theta.
csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}

STEP 7

Substitute the value of sin⁡(60∘)\sin(60^{\circ}) into the equation to calculate csc⁡(θ)\csc(\theta).
csc⁡(60∘)=1sin⁡(60∘)=132\csc(60^{\circ}) = \frac{1}{\sin(60^{\circ})} = \frac{1}{\frac{\sqrt{3}}{2}}

STEP 8

Calculate the value of csc⁡(60∘)\csc(60^{\circ}).
csc⁡(60∘)=132=23\csc(60^{\circ}) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}Therefore, if θ=60∘\theta=60^{\circ}, then sec⁡(θ)=2\sec(\theta)=2 and csc⁡(θ)=23\csc(\theta)=\frac{2}{\sqrt{3}}.

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