Math Problem Statement
Solution
We are given a right triangle, and we need to find the trigonometric functions for angle . Here’s what we know from the figure:
- The side adjacent to angle is 4.
- The side opposite angle is 14.
- The hypotenuse is unknown, but we can calculate it using the Pythagorean theorem.
Step 1: Calculate the Hypotenuse
Using the Pythagorean theorem:
Step 2: Find the Trigonometric Functions
Now, we can evaluate the six trigonometric functions for angle .
-
Sine ():
-
Cosine ():
-
Tangent ():
-
Cosecant ():
-
Secant ():
-
Cotangent ():
Final Answers:
Would you like more details on any of these calculations or further questions?
Here are some related questions to expand your understanding:
- How do you rationalize the denominator in the sine or cosine functions?
- What is the significance of each trigonometric ratio in real-world applications?
- How would the trigonometric values change if angle were a different size?
- Can the Pythagorean theorem be used to find sides in non-right triangles?
- How does the unit circle relate to these trigonometric values?
Tip: Always remember to simplify fractions and rationalize the denominator for trigonometric expressions to give exact answers.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Right Triangles
Trigonometric Functions
Formulas
Pythagorean theorem: c = √(a² + b²)
Sine: sin(θ) = opposite / hypotenuse
Cosine: cos(θ) = adjacent / hypotenuse
Tangent: tan(θ) = opposite / adjacent
Cosecant: csc(θ) = 1 / sin(θ)
Secant: sec(θ) = 1 / cos(θ)
Cotangent: cot(θ) = 1 / tan(θ)
Theorems
Pythagorean theorem
Suitable Grade Level
Grades 9-12
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