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QuestionFind sec⁡(60∘)\sec(60^{\circ}) and csc⁡(60∘)\csc(60^{\circ}) using their definitions as reciprocals of cosine and sine.

Studdy Solution

STEP 1

Assumptions1. The angle θ\theta is 60∘60^{\circ} . sec⁡(θ)\sec (\theta) is the reciprocal of cos⁡(θ)\cos (\theta)3. csc⁡(θ)\csc (\theta) is the reciprocal of sin⁡(θ)\sin (\theta)4. We are using the standard trigonometric values for an angle of 60∘60^{\circ}

STEP 2

First, we need to find the cosine of the angle θ\theta. We can do this by referring to the standard trigonometric values.
cos⁡(60∘)=12\cos (60^{\circ}) = \frac{1}{2}

STEP 3

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

STEP 4

Plug in the value for cos⁡(60∘)\cos (60^{\circ}) to calculate sec⁡(θ)\sec (\theta).
sec⁡(θ)=112\sec (\theta) = \frac{1}{\frac{1}{2}}

STEP 5

Calculate the secant of the angle θ\theta.
sec⁡(θ)=2\sec (\theta) =2

STEP 6

Next, we need to find the sine of the angle θ\theta. We can do this by referring to the standard trigonometric values.
sin⁡(60∘)=32\sin (60^{\circ}) = \frac{\sqrt{3}}{2}

STEP 7

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

STEP 8

Plug in the value for sin⁡(60∘)\sin (60^{\circ}) to calculate csc⁡(θ)\csc (\theta).
csc⁡(θ)=132\csc (\theta) = \frac{1}{\frac{\sqrt{3}}{2}}

STEP 9

Calculate the cosecant of the angle θ\theta.
csc⁡(θ)=23\csc (\theta) = \frac{2}{\sqrt{3}}So, for θ=60∘\theta=60^{\circ}, we have sec⁡(θ)=2\sec (\theta) =2 and csc⁡(θ)=23\csc (\theta) = \frac{2}{\sqrt{3}}.

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