Math Problem Statement

Use the triangle with side adjacent to angle A as 4 and side opposite angle A as 14 to evaluate each trigonometric function of angle A. Enter the exact answers.

Solution

We are given a right triangle, and we need to find the trigonometric functions for angle AA. Here’s what we know from the figure:

  • The side adjacent to angle AA is 4.
  • The side opposite angle AA is 14.
  • The hypotenuse is unknown, but we can calculate it using the Pythagorean theorem.

Step 1: Calculate the Hypotenuse

Using the Pythagorean theorem: Hypotenuse=(Adjacent side)2+(Opposite side)2\text{Hypotenuse} = \sqrt{(\text{Adjacent side})^2 + (\text{Opposite side})^2} Hypotenuse=42+142=16+196=212\text{Hypotenuse} = \sqrt{4^2 + 14^2} = \sqrt{16 + 196} = \sqrt{212} Hypotenuse=253\text{Hypotenuse} = 2\sqrt{53}

Step 2: Find the Trigonometric Functions

Now, we can evaluate the six trigonometric functions for angle AA.

  1. Sine (sinA\sin A): sinA=OppositeHypotenuse=14253=753\sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{14}{2\sqrt{53}} = \frac{7}{\sqrt{53}}

  2. Cosine (cosA\cos A): cosA=AdjacentHypotenuse=4253=253\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{2\sqrt{53}} = \frac{2}{\sqrt{53}}

  3. Tangent (tanA\tan A): tanA=OppositeAdjacent=144=72\tan A = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{14}{4} = \frac{7}{2}

  4. Cosecant (cscA\csc A): cscA=1sinA=537\csc A = \frac{1}{\sin A} = \frac{\sqrt{53}}{7}

  5. Secant (secA\sec A): secA=1cosA=532\sec A = \frac{1}{\cos A} = \frac{\sqrt{53}}{2}

  6. Cotangent (cotA\cot A): cotA=1tanA=27\cot A = \frac{1}{\tan A} = \frac{2}{7}

Final Answers:

  • sinA=753\sin A = \frac{7}{\sqrt{53}}
  • cosA=253\cos A = \frac{2}{\sqrt{53}}
  • tanA=72\tan A = \frac{7}{2}
  • cscA=537\csc A = \frac{\sqrt{53}}{7}
  • secA=532\sec A = \frac{\sqrt{53}}{2}
  • cotA=27\cot A = \frac{2}{7}

Would you like more details on any of these calculations or further questions?

Here are some related questions to expand your understanding:

  1. How do you rationalize the denominator in the sine or cosine functions?
  2. What is the significance of each trigonometric ratio in real-world applications?
  3. How would the trigonometric values change if angle AA were a different size?
  4. Can the Pythagorean theorem be used to find sides in non-right triangles?
  5. 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