Math Problem Statement

Find the angle θ in a right triangle with a hypotenuse of 47 and an adjacent side of 41.

Solution

The problem involves a right triangle where you are likely asked to find an unknown angle or side, using trigonometric relationships. In the given image, the triangle has a hypotenuse labeled as dd and a leg labeled as 41. The angle adjacent to the leg of 41 is marked as θ\theta.

To solve for θ\theta, we can use the cosine function because the cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse:

cos(θ)=adjacent sidehypotenuse=41d\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{41}{d}

Given that d=47d = 47 (as shown in the image), we can calculate θ\theta using the following steps:

cos(θ)=4147\cos(\theta) = \frac{41}{47}

First, let's calculate this value:

cos(θ)41470.87234\cos(\theta) \approx \frac{41}{47} \approx 0.87234

Next, to find θ\theta, take the arccosine (inverse cosine) of 0.87234:

θ=cos1(0.87234)29.51\theta = \cos^{-1}(0.87234) \approx 29.51^\circ

Thus, the angle θ\theta is approximately 29.5129.51^\circ.

Would you like more details or have any questions?

Here are five related questions for further exploration:

  1. What is the sine of the angle θ\theta in this triangle?
  2. How would you calculate the length of the opposite side using θ\theta and the hypotenuse?
  3. If the hypotenuse were 50 instead of 47, what would θ\theta be?
  4. Can you calculate the area of this right triangle?
  5. How does the angle θ\theta change as the adjacent side length increases?

Tip: When working with trigonometric functions, remember that the arccosine function returns an angle in degrees or radians, depending on the calculator setting. Always ensure you're interpreting the output in the correct units.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangles

Formulas

cos(θ) = adjacent/hypotenuse
θ = arccos(cos(θ))

Theorems

Trigonometric Ratios

Suitable Grade Level

Grades 9-12