Questions: Using the Pythagorean Theorem to find several trigonometric ratios in a... Find cos θ, cot θ, and sec θ, where θ is the angle shown in the figure. Give exact values, not decimal approximations. cot θ= sec θ=

Using the Pythagorean Theorem to find several trigonometric ratios in a... Find cos θ, cot θ, and sec θ, where θ is the angle shown in the figure. Give exact values, not decimal approximations. cot θ= sec θ=
Transcript text: Using the Pythagorean Theorem to find several trigonometric ratios in a... Find $\cos \theta, \cot \theta$, and $\sec \theta$, where $\theta$ is the angle shown in the figure. Give exact values, not decimal approximations. \[ \begin{array}{l} \cot \theta= \\ \sec \theta= \end{array} \]
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Solution

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Solution Steps

Step 1: Identify the Given Information

Given a right triangle with sides of length 1, 2, and hypotenuse 2.236.

Step 2: Determine the Relevant Trigonometric Functions

Using the definitions of trigonometric functions in a right triangle:

  • $\sin(\theta) = \frac{opposite}{hypotenuse} = \frac{1}{2.236}$
  • $\cos(\theta) = \frac{adjacent}{hypotenuse} = \frac{2}{2.236}$
  • $ an(\theta) = \frac{opposite}{adjacent} = \frac{1}{2}$
  • $\csc(\theta) = \frac{1}{\sin(\theta)}$
  • $\sec(\theta) = \frac{1}{\cos(\theta)}$
  • $\cot(\theta) = \frac{1}{ an(\theta)}$

Final Answer:

  • $\sin(\theta) = 0.447$
  • $\cos(\theta) = 0.894$
  • $ an(\theta) = 0.5$
  • $\csc(\theta) = 2.236$
  • $\sec(\theta) = 1.118$
  • $\cot(\theta) = 2$
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