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 (6,2)(6, 2) on the terminal side of an angle θ\theta in standard position. To find the six trigonometric functions of θ\theta, we first need to determine the radius rr, which is the distance from the origin to the point (6,2)(6, 2). This radius is given by the Pythagorean theorem:

r=x2+y2=62+22=36+4=40=210.r = \sqrt{x^2 + y^2} = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}.

Now, we can calculate the six trigonometric functions of θ\theta:

  1. Sine sinθ\sin \theta: sinθ=yr=2210=110=1010.\sin \theta = \frac{y}{r} = \frac{2}{2\sqrt{10}} = \frac{1}{\sqrt{10}} = \frac{\sqrt{10}}{10}.

  2. Cosine cosθ\cos \theta: cosθ=xr=6210=310=31010.\cos \theta = \frac{x}{r} = \frac{6}{2\sqrt{10}} = \frac{3}{\sqrt{10}} = \frac{3\sqrt{10}}{10}.

  3. Tangent tanθ\tan \theta: tanθ=yx=26=13.\tan \theta = \frac{y}{x} = \frac{2}{6} = \frac{1}{3}.

  4. Cosecant cscθ\csc \theta (the reciprocal of sine): cscθ=ry=2102=10.\csc \theta = \frac{r}{y} = \frac{2\sqrt{10}}{2} = \sqrt{10}.

  5. Secant secθ\sec \theta (the reciprocal of cosine): secθ=rx=2106=103.\sec \theta = \frac{r}{x} = \frac{2\sqrt{10}}{6} = \frac{\sqrt{10}}{3}.

  6. Cotangent cotθ\cot \theta (the reciprocal of tangent): cotθ=xy=62=3.\cot \theta = \frac{x}{y} = \frac{6}{2} = 3.

Thus, the exact values of the six trigonometric functions are:

  • sinθ=1010\sin \theta = \frac{\sqrt{10}}{10},
  • cosθ=31010\cos \theta = \frac{3\sqrt{10}}{10},
  • tanθ=13\tan \theta = \frac{1}{3},
  • cscθ=10\csc \theta = \sqrt{10},
  • secθ=103\sec \theta = \frac{\sqrt{10}}{3},
  • cotθ=3\cot \theta = 3.

Would you like any further details or clarification on any of these steps?

Here are 5 related questions to explore:

  1. How would these values change if the point were in a different quadrant?
  2. Can you derive these trigonometric functions using unit circle principles?
  3. How can the Pythagorean identity help verify these results?
  4. What are the implications of negative angles in this context?
  5. 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