Math Problem Statement
Solution
Let's solve the given problem step-by-step:
We are given:
We need to find the exact values of and .
Step 1: Identify the quadrant
Since and , we know that the angle must be in the second quadrant, where tangent is negative and cosine is negative.
Step 2: Use the Pythagorean theorem to find the hypotenuse
We can use the relationship between the sides of a right triangle:
So, let’s calculate the hypotenuse:
Step 3: Find and
In the second quadrant:
Step 4: Find
Thus:
These are the exact values for and .
Do you need any further details on this solution? Here are some related questions:
- How do we use the Pythagorean identity to solve trigonometric problems?
- Why is cosine negative in the second quadrant?
- How can we rationalize a trigonometric expression involving a square root in the denominator?
- What are the characteristics of angles in different quadrants in the unit circle?
- Can we use this approach to find other trigonometric functions for ?
Tip: Always remember to check the quadrant when dealing with trigonometric problems, as it determines the signs of the trigonometric values.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Right Triangles
Trigonometric Functions
Formulas
Pythagorean theorem
secθ = 1/cosθ
tanθ = opposite/adjacent
sinθ = opposite/hypotenuse
cosθ = adjacent/hypotenuse
Theorems
Pythagorean identity
Suitable Grade Level
Grades 10-12
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