To solve a problem involving missing sides using trigonometric functions, we can follow the steps outlined. Here’s an example problem that we can solve using these steps.
Problem: In triangle ABC, we know the following:
- Angle A = 30°
- Side a (opposite angle A) = 10
- Side b (adjacent to angle A) = x (the side we want to find)
We'll find the length of side b using the tangent function, as we have the opposite side and we want to find the adjacent side.
Steps to Solve:
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Decide which non-right angle you are going to use:
- We are using angle A (30°).
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Label the given sides based on their relationship to the angle:
- Opposite side (a) = 10 (opposite to angle A)
- Adjacent side (b) = x (the side we want to find).
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Pick which trig function you are going to use:
- We will use the tangent function because we have the opposite side and we want the adjacent side.
- Recall: \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)
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Set up your equation:
- Using the tangent function for angle A: \[ \tan(30°) = \frac{a}{b} = \frac{10}{x} \]
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Solve for x: Rearranging the equation: \[ x \cdot \tan(30°) = 10 \] \[ x = \frac{10}{\tan(30°)} \]
Now calculate \(\tan(30°)\): \(\tan(30°) = \frac{1}{\sqrt{3}} \approx 0.577\)
So substituting that back in yields: \[ x = \frac{10}{0.577} \approx 17.32 \]
- Final rounding: Round to one decimal place: \[ x \approx 17.3 \]
Conclusion: The missing side \(b\) is approximately 17.3.
Written Solution: Using the steps we've outlined:
\[ \tan(30°) = \frac{10}{x} \]
\[ \Rightarrow x \cdot \tan(30°) = 10 \] \[ \Rightarrow x = \frac{10}{\tan(30°)} \]
\[ \Rightarrow x \approx 17.3 \]
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