Question

find the value of each of the six trigonometric functions of the angle θ in the figure.

( sin\theta = square )
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed.)

( cos\theta = square )
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed.)

( \tan\theta = square )
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed.)

( csc\theta = square )
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed.)

( sec\theta = square )
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed.)

( cot\theta = square )
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using radicals as needed.)

Explanation:

To solve for the trigonometric functions of \(\theta\) in the right triangle, we first need to identify the sides relative to \(\theta\). Let's assume the triangle has:

• Hypotenuse \(= 18\)
• Opposite side to \(\theta = 9\)
• Adjacent side to \(\theta\): We can find it using the Pythagorean theorem. Let the adjacent side be \(a\). Then \(a=\sqrt{18^{2}-9^{2}}=\sqrt{324 - 81}=\sqrt{243}=9\sqrt{3}\)

1. \(\sin\theta\)

Step 1: Recall the definition of sine

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)

Step 2: Substitute the values

\(\sin\theta=\frac{9}{18}=\frac{1}{2}\)

2. \(\cos\theta\)

Step 1: Recall the definition of cosine

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)

Step 2: Substitute the values

\(\cos\theta=\frac{9\sqrt{3}}{18}=\frac{\sqrt{3}}{2}\)

3. \(\tan\theta\)

Step 1: Recall the definition of tangent

\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)

Step 2: Substitute the values

\(\tan\theta=\frac{9}{9\sqrt{3}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\) (rationalizing the denominator)

4. \(\csc\theta\)

Step 1: Recall the definition of cosecant

\(\csc\theta=\frac{1}{\sin\theta}\)

Step 2: Substitute the value of \(\sin\theta\)

\(\csc\theta=\frac{1}{\frac{1}{2}} = 2\)

5. \(\sec\theta\)

Step 1: Recall the definition of secant

\(\sec\theta=\frac{1}{\cos\theta}\)

Step 2: Substitute the value of \(\cos\theta\)

\(\sec\theta=\frac{1}{\frac{\sqrt{3}}{2}}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\) (rationalizing the denominator)

6. \(\cot\theta\)

Step 1: Recall the definition of cotangent

\(\cot\theta=\frac{1}{\tan\theta}\)

Step 2: Substitute the value of \(\tan\theta\)

\(\cot\theta=\frac{1}{\frac{\sqrt{3}}{3}}=\sqrt{3}\)

Answer:

s:
\(\sin\theta=\boldsymbol{\frac{1}{2}}\)

\(\cos\theta=\boldsymbol{\frac{\sqrt{3}}{2}}\)

\(\tan\theta=\boldsymbol{\frac{\sqrt{3}}{3}}\)

\(\csc\theta=\boldsymbol{2}\)

\(\sec\theta=\boldsymbol{\frac{2\sqrt{3}}{3}}\)

\(\cot\theta=\boldsymbol{\sqrt{3}}\)