Question

1. - / 1.5 points

find the values of sin θ, cos θ, and tan θ for the given right triangle. give the exact values.
sin θ =
cos θ =
tan θ =
right triangle with vertical leg 8, horizontal leg 7, right angle at the bottom left, and θ at the bottom right

Explanation:

Step1: Find the hypotenuse

Using the Pythagorean theorem \( c = \sqrt{a^2 + b^2} \), where \( a = 7 \) and \( b = 8 \). So \( c=\sqrt{7^2 + 8^2}=\sqrt{49 + 64}=\sqrt{113} \).

Step2: Calculate \(\sin\theta\)

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{8}{\sqrt{113}}=\frac{8\sqrt{113}}{113}\) (rationalizing the denominator).

Step3: Calculate \(\cos\theta\)

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{7}{\sqrt{113}}=\frac{7\sqrt{113}}{113}\) (rationalizing the denominator).

Step4: Calculate \(\tan\theta\)

\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{8}{7}\).

Answer:

\(\sin\theta=\frac{8\sqrt{113}}{113}\), \(\cos\theta=\frac{7\sqrt{113}}{113}\), \(\tan\theta=\frac{8}{7}\)