Math Problem Statement
Solution
To find the exact values of the remaining five trigonometric functions based on , we will use trigonometric identities and the Pythagorean theorem.
Step 1: Define the sides of the triangle
Since , we can interpret this as the opposite side being 1 and the adjacent side being 7 in a right triangle.
Step 2: Find the hypotenuse
We use the Pythagorean theorem to find the hypotenuse :
Step 3: Calculate the remaining five trigonometric functions
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Sine :
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Cosine :
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Cosecant : Cosecant is the reciprocal of sine:
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Secant : Secant is the reciprocal of cosine:
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Cotangent : Cotangent is the reciprocal of tangent:
Thus, the six trigonometric functions are:
- (given)
Let me know if you need any details or have questions!
Here are five related questions to deepen your understanding:
- How would the trigonometric functions change if was negative?
- What is the geometric interpretation of and on the unit circle?
- How do you find the trigonometric functions for angles in different quadrants?
- Can you derive the Pythagorean identity using the values of and ?
- How does the concept of reference angles apply to trigonometric functions?
Tip: The signs of the trigonometric functions change depending on the quadrant in which the angle lies, so always consider that when dealing with non-acute angles.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Right Triangles
Pythagorean Theorem
Reciprocal Trigonometric Functions
Formulas
tan(θ) = opposite / adjacent
Pythagorean Theorem: r = sqrt(opposite^2 + adjacent^2)
sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
csc(θ) = 1 / sin(θ)
sec(θ) = 1 / cos(θ)
cot(θ) = 1 / tan(θ)
Theorems
Pythagorean Theorem
Suitable Grade Level
Grades 9-12