Math Problem Statement

Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle θ, given tan(θ) = 1/7.

Solution

To find the exact values of the remaining five trigonometric functions based on tan(θ)=17\tan(\theta) = \frac{1}{7}, we will use trigonometric identities and the Pythagorean theorem.

Step 1: Define the sides of the triangle

Since tan(θ)=17\tan(\theta) = \frac{1}{7}, 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 rr: r=12+72=1+49=50=52r = \sqrt{1^2 + 7^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2}

Step 3: Calculate the remaining five trigonometric functions

  1. Sine sin(θ)\sin(\theta): sin(θ)=oppositehypotenuse=152=210\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{5\sqrt{2}} = \frac{\sqrt{2}}{10}

  2. Cosine cos(θ)\cos(\theta): cos(θ)=adjacenthypotenuse=752=7210\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{7}{5\sqrt{2}} = \frac{7\sqrt{2}}{10}

  3. Cosecant csc(θ)\csc(\theta): Cosecant is the reciprocal of sine: csc(θ)=1sin(θ)=102=52\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{10}{\sqrt{2}} = 5\sqrt{2}

  4. Secant sec(θ)\sec(\theta): Secant is the reciprocal of cosine: sec(θ)=1cos(θ)=1072=527\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{10}{7\sqrt{2}} = \frac{5\sqrt{2}}{7}

  5. Cotangent cot(θ)\cot(\theta): Cotangent is the reciprocal of tangent: cot(θ)=1tan(θ)=7\cot(\theta) = \frac{1}{\tan(\theta)} = 7

Thus, the six trigonometric functions are:

  • sin(θ)=210\sin(\theta) = \frac{\sqrt{2}}{10}
  • cos(θ)=7210\cos(\theta) = \frac{7\sqrt{2}}{10}
  • tan(θ)=17\tan(\theta) = \frac{1}{7} (given)
  • csc(θ)=52\csc(\theta) = 5\sqrt{2}
  • sec(θ)=527\sec(\theta) = \frac{5\sqrt{2}}{7}
  • cot(θ)=7\cot(\theta) = 7

Let me know if you need any details or have questions!

Here are five related questions to deepen your understanding:

  1. How would the trigonometric functions change if tan(θ)\tan(\theta) was negative?
  2. What is the geometric interpretation of sin(θ)\sin(\theta) and cos(θ)\cos(\theta) on the unit circle?
  3. How do you find the trigonometric functions for angles in different quadrants?
  4. Can you derive the Pythagorean identity using the values of sin(θ)\sin(\theta) and cos(θ)\cos(\theta)?
  5. 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