Math Problem Statement

Find the length of side 'a' given a right triangle with an angle of 69 degrees.

Solution

The image shows a right-angled triangle with one of the non-right angles labeled as 6969^\circ. The side opposite this angle is marked as aa, and we can assume that you are asked to find the length of side aa relative to the other sides or angles of the triangle.

To solve for aa, we can apply trigonometric principles. Since we have a right triangle and we know an angle (other than the right angle) and need to find a side, the sine function is appropriate because it relates the opposite side and the hypotenuse. The sine function is defined as:

sin(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

In this case:

  • θ=69\theta = 69^\circ
  • The side opposite the angle 6969^\circ is aa
  • Let the hypotenuse be denoted as cc.

So: sin(69)=ac\sin(69^\circ) = \frac{a}{c}

Thus, to find aa, we multiply both sides by cc: a=csin(69)a = c \cdot \sin(69^\circ)

To proceed further, we would need the length of the hypotenuse (cc).

Would you like more details on this, or do you know the length of the hypotenuse so we can calculate aa?

Here are 5 related questions to expand on this concept:

  1. How do we calculate the length of the adjacent side using trigonometric functions?
  2. What is the cosine function and how could it be used in this problem?
  3. How would you solve for aa if the hypotenuse were unknown?
  4. What are the other angles of the triangle if one angle is 6969^\circ?
  5. How can you apply the Pythagorean theorem to check the solution?

Tip: Always make sure the angle you are using matches the side you're solving for in trigonometric problems (opposite, adjacent, or hypotenuse).

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right Triangle
Sine Function

Formulas

sin(θ) = opposite/hypotenuse

Theorems

Trigonometric Functions

Suitable Grade Level

Grades 9-10