Problem 3 - Algebra and Trigonometry

Question

a point on the terminal side of angle \(\theta\) is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ (2,3) $$

Step-by-Step Solution

Answer

The exact values of the six trigonometric functions for \(\theta\) based on the terminal point (2,3) are as follow: \n for \(\sin \theta\), \(\cos \theta\), \(\tan \theta\), \(\csc \theta\), \(\sec \theta\), and \(\cot \theta\), respectively, we have \(\frac{3}{\sqrt{13}}\), \(\frac{2}{\sqrt{13}}\), \(\frac{3}{2}\), \(\frac{\sqrt{13}}{3}\), \(\frac{\sqrt{13}}{2}\), and \(\frac{2}{3}\).

Step 1: Calculating the hypotenuse (r)

To calculate the hypotenuse, let's use the Pythagorean Theorem with the given points (2,3), i.e. \(r = \sqrt{2^{2}+3^{2}}\). Calculate \(r\) to find the length of hypotenuse.

Step 2: Identifying coordinates

Let's identify the coordinates 'x' and 'y'. In this exercise, x = 2 and y = 3 are already given.

Step 3: Calculating six trigonometric functions

Let's calculate six trigonometric functions: \n1. For sine: \(\sin \theta = \frac{y}{r} \).2. For cosine: \(\cos \theta = \frac{x}{r} \).3. For tangent: \(\tan \theta = \frac{y}{x} \).4. For cosecant (the reciprocal of sine): \(\csc \theta = \frac{1}{\sin \theta} \).5. For secant (the reciprocal of cosine): \(\sec \theta = \frac{1}{\cos \theta} \).6. For cotangent (the reciprocal of tangent): \(\cot \theta = \frac{1}{\tan \theta} \). Substituting the values of \(x\), \(y\), and \(r\) into these formulas will yield the values of the trigonometric functions.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental concept in mathematics, particularly in geometry and trigonometry. It relates the lengths of the sides of a right triangle. According to the theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
For any right triangle, we can express this relationship with the equation:

• The hypotenuse (c) is the side opposite the right angle and is always the longest side.
• The other sides are known as the legs (a and b).
• The equation is: \( c^2 = a^2 + b^2 \).

In the context of the problem, with a point at (2,3), the x-coordinate is 2, and the y-coordinate is 3. By applying the theorem, we calculate the hypotenuse, \( r = \sqrt{2^2 + 3^2} \).
This forms the basis for further calculations of trigonometric functions.

right triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This special property sets it apart and makes it important in trigonometry. Each right triangle consists of three sides: the hypotenuse, and two legs.

• The hypotenuse is the longest side and opposite the right angle.
• The legs are the other two sides.

In our problem, the point (2,3) forms a right triangle with the origin of a coordinate plane. The x-value (2) and the y-value (3) act as the triangle’s legs, while \( r = \sqrt{13} \) is the hypotenuse.
Visualizing this triangle helps in understanding how the sides relate to each other through trigonometric functions. It is a key step in solving problems involving angles and distances in trigonometry.

sine and cosine

Sine and cosine are fundamental trigonometric functions that relate the angles of a right triangle to the ratios of its sides.

• Sine of an angle (\( \sin \theta \)) is defined as the ratio of the length of the opposite side to the hypotenuse.
• Cosine of an angle (\( \cos \theta \)) is the ratio of the length of the adjacent side to the hypotenuse.

To find the sine and cosine for the given point (2,3):

• \( \sin \theta = \frac{3}{\sqrt{13}} \)
• \( \cos \theta = \frac{2}{\sqrt{13}} \)

These functions are central in defining the angle \( \theta \), making them pivotal when determining other trigonometric values and solving related problems.

reciprocal identities

Reciprocal identities are important relationships in trigonometry that involve taking the reciprocal of basic functions. They help in expressing sin, cos, and tan in different forms. Here are the three main reciprocal identities:

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

These identities are useful in simplifying calculations and solving equations. When you know the sine, cosine, or tangent of an angle, you can easily find their reciprocals, which often appear in various trigonometric problems and solutions.