Math Problem Statement

Determine the value of z in the diagram where the angle is 54° and the adjacent side is 8 cm.

Solution

The question involves finding the length of side zz in a right triangle. The given information is as follows:

  • One angle measures 5454^\circ,
  • The adjacent side to this angle is 8 cm,
  • The side zz is the opposite side of the triangle, corresponding to the given angle.

To solve for zz, we can use the trigonometric function tangent (tan\tan) because we have the opposite side and adjacent side.

The formula for tangent is:

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

Substituting the known values:

tan(54)=z8\tan(54^\circ) = \frac{z}{8}

Solving for zz:

z=8×tan(54)z = 8 \times \tan(54^\circ)

Let me calculate the value of zz.import math

Given values

angle_degrees = 54 adjacent_side = 8

Convert angle to radians for the math.tan function

angle_radians = math.radians(angle_degrees)

Calculate the length of the opposite side (z)

z = adjacent_side * math.tan(angle_radians) zThe value of zz is approximately 11.011.0 cm, rounded to one decimal place.

Would you like more details on the steps, or do you have any further questions?

Here are some related questions to explore:

  1. How can sine and cosine be used in this triangle to find other side lengths?
  2. What are the different trigonometric ratios in a right triangle?
  3. How would the calculation change if a different angle was given instead of 54°?
  4. Can you use the Pythagorean theorem here, and if so, how?
  5. What are the common mistakes when using trigonometric functions?

Tip: Always ensure that your calculator is in the correct mode (degrees or radians) when solving trigonometric problems.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right-Angle Triangle
Tangent Function

Formulas

tan(θ) = opposite / adjacent

Theorems

Trigonometric Ratios in Right Triangles

Suitable Grade Level

Grades 9-10