Math Problem Statement
Solution
In the image, the problem involves solving a right triangle with the following information given:
- (this is one of the acute angles)
- (this is the length of one of the legs)
- is the right angle, making the hypotenuse.
You are asked to find the following:
- (the other leg of the triangle)
- (this is already known as 8)
- (the other acute angle)
Step-by-step solution:
-
Find using the tangent function:
- We know that: For , the opposite side is and the adjacent side is . So: Solving for : Using a calculator: So, .
-
Find (the hypotenuse) using the cosine function:
- The cosine of angle is: So: Using a calculator: So, .
-
Find : Since the sum of the angles in a triangle is always , and we know one angle is and , we can calculate :
Final answers:
Would you like further details or have any other questions?
Here are 5 questions you could explore based on this problem:
- How would the solution change if the angle was instead of ?
- Can you derive the same result using the Pythagorean theorem after finding one leg and the hypotenuse?
- How would you calculate the area of triangle ?
- How would you calculate the perimeter of this triangle?
- What is the sine of in this triangle, and how can it be used to verify the hypotenuse?
Tip: Always double-check your trigonometric ratios when using a calculator to avoid rounding errors.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Right Triangle
Angles
Sine, Cosine, Tangent
Formulas
tan(θ) = opposite / adjacent
cos(θ) = adjacent / hypotenuse
sin(θ) = opposite / hypotenuse
Sum of angles in a triangle = 180°
Theorems
Pythagorean Theorem
Trigonometric Ratios
Suitable Grade Level
Grades 9-11
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