Math Problem Statement

determine the exact value of each trig ratio by sketching each angle in standard position an using the related angle. The first one is tan (7pi/3)

Solution

To determine the exact value of tan(7π3)\tan\left(\frac{7\pi}{3}\right), follow these steps:


Step 1: Convert the angle into a co-terminal angle within [0,2π)[0, 2\pi)

Given 7π3\frac{7\pi}{3}, find the remainder when it is reduced by 2π2\pi.

7π32π=7π36π3=π3.\frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3}.

Thus, 7π3\frac{7\pi}{3} is co-terminal with π3\frac{\pi}{3}.


Step 2: Analyze the angle

  • The co-terminal angle π3\frac{\pi}{3} lies in the first quadrant, where tangent is positive.
  • The reference angle is π3\frac{\pi}{3} itself.

Step 3: Use the tangent ratio for π3\frac{\pi}{3}

The tangent of π3\frac{\pi}{3} is:

tan(π3)=3.\tan\left(\frac{\pi}{3}\right) = \sqrt{3}.


Step 4: Verify the quadrant

Since 7π3\frac{7\pi}{3} reduces to π3\frac{\pi}{3} in the first quadrant, where tangent is positive, the exact value is:

tan(7π3)=3.\tan\left(\frac{7\pi}{3}\right) = \sqrt{3}.


Would you like further clarification or a diagram of this process? Let me know!


Here are 5 related questions to deepen your understanding:

  1. What is the exact value of tan(11π6)\tan\left(\frac{11\pi}{6}\right)?
  2. How do co-terminal angles relate to the unit circle?
  3. Can tan(θ)\tan(\theta) ever be undefined? If so, under what conditions?
  4. How would the answer change if 7π3\frac{7\pi}{3} were in the third quadrant?
  5. What is the relationship between tangent and the other trigonometric ratios?

Tip: To find co-terminal angles, always subtract 2π2\pi (or add 2π2\pi) until the angle lies between 00 and 2π2\pi. This simplifies the calculations.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Co-terminal Angles
Unit Circle

Formulas

Co-terminal angle formula: θ - 2πk where k is an integer
Tangent ratio: tan(θ) = opposite/adjacent

Theorems

Unit Circle Trigonometric Values
Periodicity of Trigonometric Functions

Suitable Grade Level

Grades 10-12