Problem 39 - Algebra 2

Question

Find the exact values of the six trigonometric functions of \(\theta\) if the terminal side of \(\theta\) in standard position contains the given point. \((5,12)\)

Step-by-Step Solution

Answer

The six trigonometric functions are: \(\sin\theta = \frac{12}{13}\), \(\cos\theta = \frac{5}{13}\), \(\tan\theta = \frac{12}{5}\), \(\csc\theta = \frac{13}{12}\), \(\sec\theta = \frac{13}{5}\), \(\cot\theta = \frac{5}{12}\).

Step 1: Determine the Hypotenuse

Given the point (5,12), we consider it as a point on the terminal side of an angle \(\theta\). The x-coordinate is 5 and the y-coordinate is 12. We first need to find the hypotenuse \(r\) using the Pythagorean Theorem: \(r = \sqrt{x^2 + y^2}\). Substituting the values, we get \(r = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\).

Step 2: Find Sine and Cosine

The sine of \(\theta\) is given by \(\sin\theta = \frac{y}{r}\) and the cosine is given by \(\cos\theta = \frac{x}{r}\). Substituting the known values, \(\sin\theta = \frac{12}{13}\) and \(\cos\theta = \frac{5}{13}\).

Step 3: Calculate Tangent

The tangent of \(\theta\) is calculated by \(\tan\theta = \frac{y}{x}\). Substituting the given point coordinates, \(\tan\theta = \frac{12}{5}\).

Step 4: Find Cosecant, Secant, and Cotangent

The cosecant is the reciprocal of sine, \(\csc\theta = \frac{1}{\sin\theta} = \frac{13}{12}\). The secant is the reciprocal of cosine, \(\sec\theta = \frac{1}{\cos\theta} = \frac{13}{5}\). The cotangent is the reciprocal of tangent, \(\cot\theta = \frac{1}{\tan\theta} = \frac{5}{12}\).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that helps us relate the sides of a right triangle. In this exercise, the coordinates (5,12) correspond to the lengths of the legs of a right triangle, with the terminal side of angle \(\theta\) passing through this point. To find the hypotenuse or the longest side, we use the Pythagorean Theorem's formula:

• \(r = \sqrt{x^2 + y^2}\).
• This means substituting \(x = 5\) and \(y = 12\) into the formula.

So, we get \(r = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\). This calculation gives us the hypotenuse, ensuring we can continue finding trigonometric values accurately.

sine and cosine

Understanding sine and cosine is crucial for analyzing angles and their properties. They are the basic trigonometric functions,

• Sine, \(\sin\theta\), is the ratio of the opposite side to the hypotenuse.
• Cosine, \(\cos\theta\), is the ratio of the adjacent side to the hypotenuse.

Based on our earlier calculation, the hypotenuse \(r = 13\). So, \(\sin\theta = \frac{y}{r} = \frac{12}{13}\) and \(\cos\theta = \frac{x}{r} = \frac{5}{13}\). These values represent the proportions of the opposite and adjacent sides relative to the hypotenuse.

reciprocal identities

Reciprocal identities are handy for finding the remaining trigonometric functions. Once you know sine, cosine, and tangent, you can easily determine cosecant, secant, and cotangent:

• Cosecant is the reciprocal of sine, \(\csc\theta = \frac{1}{\sin\theta} = \frac{13}{12}\).
• Secant is the reciprocal of cosine, \(\sec\theta = \frac{1}{\cos\theta} = \frac{13}{5}\).
• Cotangent is the reciprocal of tangent, \(\cot\theta = \frac{1}{\tan\theta} = \frac{5}{12}\).

These identities are simply alternate ways of expressing the fundamental trigonometric functions. They are invaluable tools across many fields, from solving triangles to modeling periodic phenomena.