What is the exact value of tan 60 degrees?
Understand the Problem
The question is asking for the exact value of the tangent of 60 degrees, which is a trigonometric value.
Answer
The exact value of the tangent of 60 degrees is $ \sqrt{3} $.
Answer for screen readers
The exact value of the tangent of 60 degrees is $ \sqrt{3} $.
Steps to Solve
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Identify the relationship in the right triangle The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. For a 60-degree angle, we can use the known values from a special right triangle.
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Use the 30-60-90 triangle values In a 30-60-90 triangle, the ratios of the lengths of the sides are well-known:
- The side opposite the 30-degree angle (the shorter leg) is ( x ).
- The side opposite the 60-degree angle (the longer leg) is ( x\sqrt{3} ).
- The hypotenuse is ( 2x ).
- Calculate the tangent of 60 degrees The tangent of 60 degrees can be expressed as: $$ \tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} $$ For the 60-degree angle: $$ \tan(60^\circ) = \frac{x\sqrt{3}}{x} = \sqrt{3} $$
The exact value of the tangent of 60 degrees is $ \sqrt{3} $.
More Information
The tangent function is one of the basic functions in trigonometry, and knowing the values for standard angles like 30, 45, and 60 degrees can help in many mathematical applications. The value $\sqrt{3}$ is approximately 1.732, which is useful in various calculations.
Tips
- Forgetting to use the correct triangle ratios: Ensure you use the 30-60-90 triangle ratios correctly when computing values.
- Confusing the tangent function with sine or cosine: Remember, tangent is the ratio of opposite to adjacent, while sine and cosine have different definitions.
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