Math Problem Statement

In the right triangle with sides 41.76, 22, and hypotenuse 47.2, what is the value of angle θ?

Solution

The image shows a right triangle with the following dimensions:

  • Opposite side to angle θ\theta: 41.76
  • Adjacent side to angle θ\theta: 22
  • Hypotenuse: 47.2

We can use trigonometric ratios to solve for angle θ\theta. Since the triangle is a right triangle, we can apply the following relationships:

  1. Tangent (tan) of an angle is the ratio of the opposite side to the adjacent side: tan(θ)=OppositeAdjacent=41.7622\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{41.76}{22}

  2. To find θ\theta, take the arctangent (inverse of tangent): θ=tan1(41.7622)\theta = \tan^{-1} \left( \frac{41.76}{22} \right)

Let me calculate θ\theta:

θtan1(1.898)62.63\theta \approx \tan^{-1} \left( 1.898 \right) \approx 62.63^\circ

Thus, the angle θ62.63\theta \approx 62.63^\circ.

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

  1. What is the sine of angle θ\theta using the triangle's sides?
  2. How can we calculate the cosine of angle θ\theta?
  3. Can the Pythagorean theorem verify the given side lengths of the triangle?
  4. How would you find the area of this right triangle?
  5. How can we express angle θ\theta in radians?

Tip: When working with right triangles, the Pythagorean theorem is a reliable way to check the accuracy of the given side lengths.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangles
Inverse Trigonometric Functions

Formulas

tan(θ) = Opposite / Adjacent = 41.76 / 22
θ = tan^(-1)(41.76 / 22)

Theorems

Pythagorean Theorem
Trigonometric Ratios

Suitable Grade Level

Grades 10-12