Solving with the tangent (tan) function generally involves using it to find a missing side length or angle in a right triangle. The tangent function relates an angle to the ratio of the length of the opposite side to the length of the adjacent side. Here's a breakdown:
Understanding Tangent
The tangent of an angle (often written as tan(θ)) in a right triangle is defined as:
tan(θ) = Opposite / Adjacent
- Opposite: The length of the side opposite to the angle θ.
- Adjacent: The length of the side adjacent to the angle θ (not the hypotenuse).
Solving for a Missing Side
If you know the angle θ and the length of one of the sides (opposite or adjacent), you can solve for the other side.
Example:
Suppose you have a right triangle where:
- Angle θ = 36 degrees
- Adjacent side = 7
You want to find the length of the opposite side (let's call it x).
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Set up the equation:
tan(36°) = x / 7
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Solve for x:
x = 7 * tan(36°)
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Calculate using a calculator:
x ≈ 7 * 0.7265 ≈ 5.09
Therefore, the length of the opposite side is approximately 5.09.
Solving for a Missing Angle
If you know the lengths of the opposite and adjacent sides, you can solve for the angle θ using the inverse tangent function (arctan or tan-1).
Example:
Suppose you have a right triangle where:
- Opposite side = 5
- Adjacent side = 7
You want to find the angle θ.
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Set up the equation:
tan(θ) = 5 / 7
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Solve for θ:
θ = tan-1(5 / 7)
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Calculate using a calculator:
θ ≈ 35.54 degrees
Therefore, the angle θ is approximately 35.54 degrees.
Steps for Solving with Tan
- Identify the right triangle: Make sure the problem involves a right triangle.
- Identify the known values: Determine which angles and side lengths are given.
- Determine what you need to find: Identify the missing side length or angle.
- Set up the equation: Use the tangent formula (tan(θ) = Opposite / Adjacent) or its inverse (θ = tan-1(Opposite / Adjacent)).
- Solve the equation: Use algebra to isolate the unknown variable.
- Calculate the answer: Use a calculator to find the numerical value.
