Question

use the pythagorean theorem to find the length of the missing side of the right triangle. then find the value of each of the six trigonometric functions of θ. the length of the missing side of the right triangle is c = \square. \sin \theta = \square (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) \cos \theta = \square (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) \tan \theta = \square (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) \csc \theta = \square (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) \sec \theta = \square (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) \cot \theta = \square (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find the hypotenuse \( c \)

In a right triangle, the Pythagorean theorem is \( a^2 + b^2 = c^2 \). Given \( a = 7 \) and \( b = 24 \), we substitute these values:
\( 7^2 + 24^2 = c^2 \)
\( 49 + 576 = c^2 \)
\( 625 = c^2 \)
Taking the square root of both sides, \( c = \sqrt{625} = 25 \).

Step2: Find \( \sin\theta \)

The sine of an angle \( \theta \) in a right triangle is \( \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \). For \( \theta \), the opposite side is \( a = 7 \) and the hypotenuse is \( c = 25 \), so:
\( \sin\theta = \frac{7}{25} \)

Step3: Find \( \cos\theta \)

The cosine of an angle \( \theta \) is \( \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \). The adjacent side to \( \theta \) is \( b = 24 \) and the hypotenuse is \( 25 \), so:
\( \cos\theta = \frac{24}{25} \)

Step4: Find \( \tan\theta \)

The tangent of an angle \( \theta \) is \( \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \). The opposite side is \( 7 \) and the adjacent side is \( 24 \), so:
\( \tan\theta = \frac{7}{24} \)

Step5: Find \( \csc\theta \)

The cosecant is the reciprocal of sine, so \( \csc\theta = \frac{1}{\sin\theta} \). Since \( \sin\theta = \frac{7}{25} \), we have:
\( \csc\theta = \frac{25}{7} \)

Step6: Find \( \sec\theta \)

The secant is the reciprocal of cosine, so \( \sec\theta = \frac{1}{\cos\theta} \). Since \( \cos\theta = \frac{24}{25} \), we have:
\( \sec\theta = \frac{25}{24} \)

Step7: Find \( \cot\theta \)

The cotangent is the reciprocal of tangent, so \( \cot\theta = \frac{1}{\tan\theta} \). Since \( \tan\theta = \frac{7}{24} \), we have:
\( \cot\theta = \frac{24}{7} \)

Answer:

• Missing side \( c = 25 \)
• \( \sin\theta = \frac{7}{25} \)
• \( \cos\theta = \frac{24}{25} \)
• \( \tan\theta = \frac{7}{24} \)
• \( \csc\theta = \frac{25}{7} \)
• \( \sec\theta = \frac{25}{24} \)
• \( \cot\theta = \frac{24}{7} \)