Question

for the triangle shown in the figure below what are each of the following? (let y = 84.0 m and r = 91.0 m. assume the triangle is a right triangle.) (a) the length of the unknown side x 35.0 m (b) the tangent of θ enter a number. θ is the ratio of two sides of the triangle, but you need to make sure you are expressing the ratio correctly. (c) the sin of φ 0.385

Explanation:

Part (a): Length of side \( x \)

Step 1: Identify the triangle type

The triangle is a right triangle, so we can use the Pythagorean theorem: \( r^2 = x^2 + y^2 \), where \( r \) is the hypotenuse, and \( x, y \) are the legs. We need to solve for \( x \).

Step 2: Rearrange the Pythagorean theorem

From \( r^2 = x^2 + y^2 \), we get \( x^2 = r^2 - y^2 \). Substitute \( r = 91.0 \, \text{m} \) and \( y = 84.0 \, \text{m} \):
\[
x^2 = (91.0)^2 - (84.0)^2
\]

Step 3: Calculate the squares

\[
(91.0)^2 = 8281, \quad (84.0)^2 = 7056
\]

Step 4: Subtract the squares

\[
x^2 = 8281 - 7056 = 1225
\]

Step 5: Take the square root

\[
x = \sqrt{1225} = 35.0 \, \text{m}
\]

Part (b): Tangent of \( \theta \)

Step 1: Recall the definition of tangent

In a right triangle, \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \) to angle \( \theta \). For angle \( \theta \), the opposite side is \( x \) and the adjacent side is \( y \).

Step 2: Substitute the values

\( x = 35.0 \, \text{m} \) and \( y = 84.0 \, \text{m} \), so:
\[
\tan(\theta) = \frac{x}{y} = \frac{35.0}{84.0}
\]

Step 3: Simplify the fraction

\[
\frac{35.0}{84.0} = \frac{5}{12} \approx 0.4167
\]

Part (c): Sine of \( \phi \)

Step 1: Recall the definition of sine

In a right triangle, \( \sin(\phi) = \frac{\text{opposite}}{\text{hypotenuse}} \) to angle \( \phi \). For angle \( \phi \), the opposite side is \( x \) and the hypotenuse is \( r \).

Step 2: Substitute the values

\( x = 35.0 \, \text{m} \) and \( r = 91.0 \, \text{m} \), so:
\[
\sin(\phi) = \frac{x}{r} = \frac{35.0}{91.0}
\]

Step 3: Simplify the fraction

\[
\frac{35.0}{91.0} = \frac{5}{13} \approx 0.3846 \approx 0.385
\]

Answer:

s:
(a) \( \boldsymbol{35.0 \, \text{m}} \)
(b) \( \boldsymbol{\approx 0.417} \) (or \( \frac{5}{12} \))
(c) \( \boldsymbol{\approx 0.385} \)