Math Problem Statement
find the exact value of each of the six trigonometric functions of theta, if (6, 2) is a point on the terminal side of angle theta.
Solution
We are given a point on the terminal side of an angle in standard position. To find the six trigonometric functions of , we first need to determine the radius , which is the distance from the origin to the point . This radius is given by the Pythagorean theorem:
Now, we can calculate the six trigonometric functions of :
-
Sine :
-
Cosine :
-
Tangent :
-
Cosecant (the reciprocal of sine):
-
Secant (the reciprocal of cosine):
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Cotangent (the reciprocal of tangent):
Thus, the exact values of the six trigonometric functions are:
- ,
- ,
- ,
- ,
- ,
- .
Would you like any further details or clarification on any of these steps?
Here are 5 related questions to explore:
- How would these values change if the point were in a different quadrant?
- Can you derive these trigonometric functions using unit circle principles?
- How can the Pythagorean identity help verify these results?
- What are the implications of negative angles in this context?
- How do trigonometric functions relate to real-world applications like navigation?
Tip: The reciprocal identities are a quick way to check your work when finding cosecant, secant, and cotangent!
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Cartesian Coordinates
Reciprocal Trigonometric Functions
Pythagorean Theorem
Formulas
r = √(x² + y²)
sin θ = y / r
cos θ = x / r
tan θ = y / x
csc θ = r / y
sec θ = r / x
cot θ = x / y
Theorems
Pythagorean Theorem
Reciprocal Identities
Suitable Grade Level
Grades 9-12
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