Problem 25 - Algebra and Trigonometry Real Mathematics, Real People

Question

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$(-3,-\sqrt{7})$$

Step-by-Step Solution

Answer

The exact values of the six trigonometric functions for the angle where the point (-3,-\sqrt{7}) lies on the terminal side are: Sin\(\theta\) = -\sqrt{7}/4, Cos\(\theta\) = -3/4, Tan\(\theta\) = \sqrt{7}/3, Cosec\(\theta\) = -4/\sqrt{7}, Sec\(\theta\) = -4/3, Cot\(\theta\) = 3/\sqrt{7}

Step 1: Calculate the radius \(r\)

First, calculate the radius or hypotenuse of the triangle formed by \(x\), \(y\), and \(r\). We can use the Pythagorean theorem to calculate \(r\). The formula is \(r = \sqrt{x^2+y^2}\). Here, \(x=-3\) and \(y=-\sqrt{7}\). So, \(r = \sqrt{(-3)^2+(-\sqrt{7})^2} = \sqrt{9+7} = \sqrt{16} = 4.\) So, \(r\) equals to 4.

Step 2: Calculate the six trigonometric functions

Now use \(x\), \(y\), and \(r\) to calculate the exact value of the six trigonometric functions. In this case: \n1. Sin\(\theta\) = \(y/r = -\sqrt{7}/4\)\n2. Cos\(\theta\) = \(x/r = -3/4\)\n3. Tan\(\theta\) = \(y/x = \sqrt{7}/3\)\n4. Cosec\(\theta\) is the reciprocal of Sin\(\theta\) = \(-4/\sqrt{7}\)\n5. Sec\(\theta\) is the reciprocal of Cos\(\theta\) = \(-4/3\)\n6. Cot\(\theta\) is the reciprocal of Tan\(\theta\) = \(3/\sqrt{7}\)

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics, especially in trigonometry. It relates the lengths of the sides of a right triangle. By the theorem, the square of the hypotenuse (the side opposite the right angle) is the sum of the squares of the other two sides.

For a point \(x, y\) on the terminal side in standard position, we use the coordinates as the legs of a right triangle. Therefore, the hypotenuse \(r\) can be calculated by \[ r = \sqrt{x^2 + y^2} \]. Here, the values given are \(x = -3\) and \(y = -\sqrt{7}\).

Plug into the formula:

• \( r = \sqrt{(-3)^2 + (-\sqrt{7})^2} = \sqrt{9 + 7} = \sqrt{16} = 4 \)

This computation is crucial for determining other trigonometric functions.

Standard Position

Angles in trigonometry are often described in standard position, which means the angle's vertex is at the origin of a coordinate plane, and its initial side lies along the positive x-axis.

This setup helps in defining trigonometric functions using coordinates. When the terminal side of the angle passes through a point like \(-3, -\sqrt{7}\), it forms a triangle with the x-axis and helps determine key values.

Moving to coordinates is a common strategy to visualize angles and makes calculating trigonometric functions possible using the point's location.

Reciprocal Identities

Reciprocal identities are essential in trigonometry, providing an easy way to find three additional functions: cosecant, secant, and cotangent.

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

For the given point, we had calculated:

• \( \text{csc}(\theta) = -\frac{4}{\sqrt{7}} \)
• \( \text{sec}(\theta) = -\frac{4}{3} \)
• \( \text{cot}(\theta) = \frac{3}{\sqrt{7}} \)

These identities offer a straightforward method to extend the range of functions without direct division.

Exact Values

Finding the exact values of trigonometric functions is crucial for precise calculations. These values stem from specific circle points or basic angles.

In our exercise, calculating the functions for the point \(-3, -\sqrt{7}\), with \(x, y,\) and \(r = 4\), we derived:

• \( \sin(\theta) = -\frac{\sqrt{7}}{4} \)
• \( \cos(\theta) = -\frac{3}{4} \)
• \( \tan(\theta) = \frac{\sqrt{7}}{3} \)

Working with exact values helps avoid rounding errors, providing precise outputs necessary for mathematical proofs and applications.

Accurate calculation of these values assures consistent results in various trigonometric scenarios.