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Solved on Mar 20, 2024

Find exact values for sec⁡(θ)\sec (\theta), csc⁡(θ)\csc (\theta), tan⁡(θ)\tan (\theta), and cot⁡(θ)\cot (\theta) when θ=5π4\theta=\frac{5 \pi}{4}.

STEP 1

1. The trigonometric functions secant (sec⁡\sec), cosecant (csc⁡\csc), tangent (tan⁡\tan), and cotangent (cot⁡\cot) are the reciprocals of cosine (cos⁡\cos), sine (sin⁡\sin), and their respective reciprocals.
2. The angle θ=5π4\theta = \frac{5\pi}{4} is in the third quadrant of the unit circle, where both sine and cosine are negative.
3. The exact values of sine and cosine for the commonly used angles can be determined using the unit circle or trigonometric identities.
4. The values of tan⁡(θ)\tan(\theta) and cot⁡(θ)\cot(\theta) can be found using the relationship tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} and cot⁡(θ)=1tan⁡(θ)\cot(\theta) = \frac{1}{\tan(\theta)} respectively.

STEP 2

1. Determine the values of cos⁡(θ)\cos(\theta) and sin⁡(θ)\sin(\theta) for θ=5π4\theta = \frac{5\pi}{4}.
2. Calculate sec⁡(θ)\sec(\theta) and csc⁡(θ)\csc(\theta) using their definitions as reciprocals of cos⁡(θ)\cos(\theta) and sin⁡(θ)\sin(\theta).
3. Calculate tan⁡(θ)\tan(\theta) using the relationship with sin⁡(θ)\sin(\theta) and cos⁡(θ)\cos(\theta).
4. Calculate cot⁡(θ)\cot(\theta) as the reciprocal of tan⁡(θ)\tan(\theta).

STEP 3

Find the value of cos⁡(θ)\cos(\theta) for θ=5π4\theta = \frac{5\pi}{4}.
cos⁡(5π4)=−22 \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}

STEP 4

Find the value of sin⁡(θ)\sin(\theta) for θ=5π4\theta = \frac{5\pi}{4}.
sin⁡(5π4)=−22 \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}

STEP 5

Calculate sec⁡(θ)\sec(\theta) as the reciprocal of cos⁡(θ)\cos(\theta).
sec⁡(5π4)=1cos⁡(5π4)=1−22=−2 \sec\left(\frac{5\pi}{4}\right) = \frac{1}{\cos\left(\frac{5\pi}{4}\right)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}

STEP 6

Calculate csc⁡(θ)\csc(\theta) as the reciprocal of sin⁡(θ)\sin(\theta).
csc⁡(5π4)=1sin⁡(5π4)=1−22=−2 \csc\left(\frac{5\pi}{4}\right) = \frac{1}{\sin\left(\frac{5\pi}{4}\right)} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2}

STEP 7

Calculate tan⁡(θ)\tan(\theta) using the relationship with sin⁡(θ)\sin(\theta) and cos⁡(θ)\cos(\theta).
tan⁡(5π4)=sin⁡(5π4)cos⁡(5π4)=−22−22=1 \tan\left(\frac{5\pi}{4}\right) = \frac{\sin\left(\frac{5\pi}{4}\right)}{\cos\left(\frac{5\pi}{4}\right)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1

STEP 8

Calculate cot⁡(θ)\cot(\theta) as the reciprocal of tan⁡(θ)\tan(\theta).
cot⁡(5π4)=1tan⁡(5π4)=11=1 \cot\left(\frac{5\pi}{4}\right) = \frac{1}{\tan\left(\frac{5\pi}{4}\right)} = \frac{1}{1} = 1
The exact values for the trigonometric functions at θ=5π4\theta = \frac{5\pi}{4} are: sec⁡(θ)=−2\sec(\theta) = -\sqrt{2} csc⁡(θ)=−2\csc(\theta) = -\sqrt{2} tan⁡(θ)=1\tan(\theta) = 1 cot⁡(θ)=1\cot(\theta) = 1

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