Solving for the tangent often depends on what information you have and what you're trying to find. The tangent function (tan) relates the angle of a right triangle to the ratio of the length of the opposite side to the adjacent side. Here's a breakdown of how to solve for the tangent in different scenarios:
1. Finding the Tangent of an Angle
If you know the angle, you can find the tangent value using a calculator or trigonometric tables.
- Calculator: Ensure your calculator is in the correct mode (degrees or radians). Input the angle and press the "tan" button.
- Example: tan(35°) ≈ 0.7002
2. Solving for a Side Length in a Right Triangle Using Tangent
If you know one side length and an angle (other than the right angle) in a right triangle, you can use the tangent function to find the other side.
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Formula: tan(angle) = Opposite / Adjacent
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Example: Suppose you have a right triangle where the angle is 35°, the adjacent side is 17, and you need to find the opposite side (let's call it 'a').
- tan(35°) = a / 17
- a = 17 * tan(35°)
- a ≈ 17 * 0.7002
- a ≈ 11.90
3. Finding the Angle Using the Inverse Tangent (arctan or tan-1)
If you know the lengths of the opposite and adjacent sides, you can use the inverse tangent function to find the angle.
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Formula: angle = tan-1 (Opposite / Adjacent)
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Calculator: Use the "arctan" or "tan-1" function on your calculator (usually accessed by pressing "shift" or "2nd" then "tan").
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Example: If the opposite side is 12 and the adjacent side is 5:
- angle = tan-1 (12 / 5)
- angle ≈ tan-1 (2.4)
- angle ≈ 67.38°
Summary
In essence, solving for the tangent involves using the tangent function (tan), its inverse (arctan or tan-1), and the relationship between angles and side lengths in a right triangle.
