Consider a right angled triangle where one of the non-hypotenuse sides has length 3 and the angle it makes with the hypotenuse is 65 degrees. What is the area of this triangle?
Understand the Problem
The question describes a right-angled triangle and asks for its area. We are given the length of one of the non-hypotenuse sides (which is a leg) and the angle between that side and the hypotenuse. We need to use trigonometry to find the length of the other leg, and then calculate the area using the formula (1/2) * base * height.
Answer
$\frac{25}{2}(2 - \sqrt{3})$
Answer for screen readers
$\frac{25}{2}(2 - \sqrt{3})$
Steps to Solve
- Identify the knowns and unknowns
We know one leg of the right triangle, let's call it $a = 5$. We also know the angle between this leg and the hypotenuse, $\theta = \frac{\pi}{12}$. We need to find the other leg, which we'll call $b$, to calculate the area.
- Use trigonometry to find the other leg
Since we know the adjacent side ($a$) and the angle $\theta$, we can use the tangent function to find the opposite side ($b$):
$$ \tan(\theta) = \frac{b}{a} $$
Therefore:
$$ b = a \cdot \tan(\theta) $$
- Substitute the given values
Substitute $a = 5$ and $\theta = \frac{\pi}{12}$ into the equation:
$$ b = 5 \cdot \tan\left(\frac{\pi}{12}\right) $$
- Calculate $\tan(\frac{\pi}{12})$
We know that $\tan(\frac{\pi}{12}) = 2 - \sqrt{3}$. Therefore,
$$ b = 5 \cdot (2 - \sqrt{3}) $$
- Calculate the area of the triangle
The area of a right triangle is given by:
$$ \text{Area} = \frac{1}{2} \cdot a \cdot b $$
Substitute $a = 5$ and $b = 5(2 - \sqrt{3})$:
$$ \text{Area} = \frac{1}{2} \cdot 5 \cdot 5(2 - \sqrt{3}) $$
$$ \text{Area} = \frac{25}{2} (2 - \sqrt{3}) $$
$\frac{25}{2}(2 - \sqrt{3})$
More Information
The exact area of the right triangle is $\frac{25}{2}(2 - \sqrt{3})$. We used trigonometry (tangent function) to find the length of the missing leg and then applied the standard formula for the area of a right triangle.
Tips
A common mistake is using the incorrect trigonometric function (e.g., sine or cosine instead of tangent). It is important to correctly identify which sides are adjacent and opposite to the given angle. Another error could occur when evaluating $\tan(\frac{\pi}{12})$. Remembering trigonometric values for common angles can help prevent mistakes.
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