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Part C: Trigonometry and Geometry

Solve the following using SOH, CAH, TOA, and the Pythagorean Theorem for right triangles.

1. In a right triangle, the leg adjacent to an angle of [tex]\(23^\circ\)[/tex] is 12 cm long. How long is the leg opposite the [tex]\(23^\circ\)[/tex] angle, to 2 decimal places?



Answer :

GinnyAnswer
Sure, let's solve the given problem step by step using trigonometric principles. We need to find the length of the leg opposite to the 23° angle in a right triangle where the adjacent leg is 12 cm long. We'll use the tangent function from trigonometry, which relates the opposite leg to the adjacent leg and the angle. Here is the detailed process:

1. Understand Tangent Function (TOA)
- The tangent of an angle in a right triangle is the ratio of the length of the opposite leg to the length of the adjacent leg.
- This can be written as: [tex]\(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)[/tex]

2. Identify Known Values
- Angle [tex]\(\theta\)[/tex] = 23°
- Adjacent leg = 12 cm

3. Set Up the Equation
- Using the tangent function: [tex]\(\tan(23°) = \frac{\text{opposite}}{12}\)[/tex]

4. Isolate the Opposite Leg
- Multiply both sides by the length of the adjacent leg to solve for the opposite leg:
[tex]\[ \text{opposite} = 12 \times \tan(23°) \][/tex]

5. Calculate the Tangent
- Find the tangent of 23°. Using the tangent value directly gives:
[tex]\[ \tan(23°) \approx 0.424474816209604 \][/tex]
(Note: The precise value is used here for explanation purposes. In an actual setting, you'd use a calculator.)

6. Calculate the Length of Opposite Leg
- Substitute the tangent value back into the equation:
[tex]\[ \text{opposite} = 12 \times 0.424474816209604 \approx 5.093697794515257 \][/tex]

7. Round the Result
- Round the length of the opposite leg to 2 decimal places:
[tex]\[ \text{The length of the opposite leg} \approx 5.09 \text{ cm} \][/tex]

Hence, the length of the leg opposite the 23° angle in the right triangle is approximately 5.09 cm.