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 , follow these steps:
Step 1: Convert the angle into a co-terminal angle within
Given , find the remainder when it is reduced by .
Thus, is co-terminal with .
Step 2: Analyze the angle
- The co-terminal angle lies in the first quadrant, where tangent is positive.
- The reference angle is itself.
Step 3: Use the tangent ratio for
The tangent of is:
Step 4: Verify the quadrant
Since reduces to in the first quadrant, where tangent is positive, the exact value is:
Would you like further clarification or a diagram of this process? Let me know!
Here are 5 related questions to deepen your understanding:
- What is the exact value of ?
- How do co-terminal angles relate to the unit circle?
- Can ever be undefined? If so, under what conditions?
- How would the answer change if were in the third quadrant?
- What is the relationship between tangent and the other trigonometric ratios?
Tip: To find co-terminal angles, always subtract (or add ) until the angle lies between and . 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