Questions: A right triangle with acute angle θ is given. Evaluate the six trigonometric functions of the acute angle θ.

A right triangle with acute angle θ is given. Evaluate the six trigonometric functions of the acute angle θ.
Transcript text: A right triangle with acute angle $\theta$ is given. Evaluate the six trigonometric functions of the acute angle $\theta$.
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Solution

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Solution Steps

To evaluate the six trigonometric functions of an acute angle θ\theta in a right triangle, we need to know the lengths of the sides of the triangle. Let's assume the lengths of the opposite side, adjacent side, and hypotenuse are given as opposite, adjacent, and hypotenuse, respectively. The six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent, which can be calculated using these side lengths.

Step 1: Identify the Side Lengths

In a right triangle, we are given the side lengths: the opposite side is 3, the adjacent side is 4, and the hypotenuse is 5.

Step 2: Calculate the Sine of θ\theta

The sine of angle θ\theta is calculated as the ratio of the length of the opposite side to the hypotenuse: sin(θ)=oppositehypotenuse=35=0.6000 \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} = 0.6000

Step 3: Calculate the Cosine of θ\theta

The cosine of angle θ\theta is calculated as the ratio of the length of the adjacent side to the hypotenuse: cos(θ)=adjacenthypotenuse=45=0.8000 \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} = 0.8000

Step 4: Calculate the Tangent of θ\theta

The tangent of angle θ\theta is calculated as the ratio of the length of the opposite side to the adjacent side: tan(θ)=oppositeadjacent=34=0.7500 \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4} = 0.7500

Step 5: Calculate the Cosecant of θ\theta

The cosecant of angle θ\theta is the reciprocal of the sine: csc(θ)=1sin(θ)=531.6667 \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{5}{3} \approx 1.6667

Step 6: Calculate the Secant of θ\theta

The secant of angle θ\theta is the reciprocal of the cosine: sec(θ)=1cos(θ)=54=1.2500 \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{5}{4} = 1.2500

Step 7: Calculate the Cotangent of θ\theta

The cotangent of angle θ\theta is the reciprocal of the tangent: cot(θ)=1tan(θ)=431.3333 \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{4}{3} \approx 1.3333

Final Answer

sin(θ)=35,cos(θ)=45,tan(θ)=34,csc(θ)=53,sec(θ)=54,cot(θ)=43 \sin(\theta) = \frac{3}{5}, \quad \cos(\theta) = \frac{4}{5}, \quad \tan(\theta) = \frac{3}{4}, \quad \csc(\theta) = \frac{5}{3}, \quad \sec(\theta) = \frac{5}{4}, \quad \cot(\theta) = \frac{4}{3}

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