What is the area of a right triangle when the hypotenuse is 5 cm long and one angle is π/10?
Understand the Problem
The question is asking for the area of a right triangle given the length of the hypotenuse and one angle. To solve it, we will use the formula for the area of a triangle, which is 0.5 * base * height. We need to find the base and height using trigonometric functions based on the given angle.
Answer
The area of the triangle is \( A = \frac{25}{4} \cdot \sin(20^\circ) \, \text{cm}^2 \).
Answer for screen readers
The area of the triangle is given by: $$ A = \frac{25}{4} \cdot \sin(20^\circ) , \text{cm}^2 $$
Steps to Solve
- Identify the sides of the triangle using trigonometric ratios
Given the hypotenuse ( c = 5 , \text{cm} ) and angle ( \theta = 10^\circ ), we can find the two legs (base and height) of the triangle using the sine and cosine functions:
- Base ( b = c \cdot \cos(\theta) )
- Height ( h = c \cdot \sin(\theta) )
- Calculate the base using the cosine function
To find the base: $$ b = 5 \cdot \cos(10^\circ) $$
- Calculate the height using the sine function
To find the height: $$ h = 5 \cdot \sin(10^\circ) $$
- Calculate the area of the triangle
The area ( A ) of the triangle can be calculated using the area formula for a triangle: $$ A = \frac{1}{2} \cdot b \cdot h $$ Substituting the expressions we derived for base and height, the equation becomes: $$ A = \frac{1}{2} \cdot (5 \cdot \cos(10^\circ)) \cdot (5 \cdot \sin(10^\circ)) $$
- Simplify the area expression
Combine the terms: $$ A = \frac{25}{2} \cdot \cos(10^\circ) \cdot \sin(10^\circ) $$
Utilizing the identity ( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) ), we can rewrite the expression: $$ A = \frac{25}{4} \cdot \sin(20^\circ) $$
The area of the triangle is given by: $$ A = \frac{25}{4} \cdot \sin(20^\circ) , \text{cm}^2 $$
More Information
The area of the triangle can vary depending on the angle given. Understanding trigonometric functions is crucial for solving problems involving triangles with angles and hypotenuse lengths.
Tips
- Forgetting to convert angles from degrees to radians if necessary. Always check the mode of your calculator!
- Mixing up sine and cosine functions. Remember that sine is associated with the opposite side and cosine with the adjacent side.
AI-generated content may contain errors. Please verify critical information
