Problem 71 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact values of the six trigonometric functions of \(\theta\) if \(\theta\) is in standard position and \(P\) is on the terminal side. $$P(4,-3)$$

Step-by-Step Solution

Answer

The exact values are: \(\sin \theta = -\frac{3}{5}\), \(\cos \theta = \frac{4}{5}\), \(\tan \theta = -\frac{3}{4}\), \(\csc \theta = -\frac{5}{3}\), \(\sec \theta = \frac{5}{4}\), \(\cot \theta = -\frac{4}{3}\).

Step 1: Determine Radius 'r'

To find the trigonometric functions, start by determining the hypotenuse (or 'r') of the right triangle formed by the point \( P(4, -3) \). The formula for 'r' in terms of x and y is \[ r = \sqrt{x^2 + y^2} \]Plug in the values:\[ r = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

Step 2: Calculate \\(\sin \theta\)

The sine of angle \(\theta\) is defined as the ratio of the opposite side to the hypotenuse. Using the point (4,-3):\[ \sin \theta = \frac{y}{r} = \frac{-3}{5} \]

Step 3: Calculate \\(\cos \theta\)

The cosine of angle \(\theta\) is the ratio of the adjacent side to the hypotenuse:\[ \cos \theta = \frac{x}{r} = \frac{4}{5} \]

Step 4: Calculate \\(\tan \theta\)

The tangent of angle \(\theta\) is the ratio of the opposite side to the adjacent side:\[ \tan \theta = \frac{y}{x} = \frac{-3}{4} \]

Step 5: Calculate \\(\csc \theta\)

The cosecant is the reciprocal of the sine function:\[ \csc \theta = \frac{1}{\sin \theta} = \frac{5}{-3} = -\frac{5}{3} \]

Step 6: Calculate \\(\sec \theta\)

The secant is the reciprocal of the cosine function:\[ \sec \theta = \frac{1}{\cos \theta} = \frac{5}{4} \]

Step 7: Calculate \\(\cot \theta\)

The cotangent is the reciprocal of the tangent function:\[ \cot \theta = \frac{1}{\tan \theta} = \frac{4}{-3} = -\frac{4}{3} \]

Key Concepts

Sine

The sine function, often abbreviated as "sin," is a fundamental trigonometric function used to relate the angles of a right triangle to the lengths of its sides. In the context of a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse, which is the longest side in a right triangle.
For an angle \(\theta\), we symbolize it as follows:

• \(\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\)

In practical terms, if you have a triangle with a hypotenuse of length 5 and the side opposite \(\theta\) is 3, then \(\sin \theta = \frac{3}{5}\). It's crucial to remember that the hypotenuse is always the longest side, and this ratio helps calculate the size of angle \(\theta\).

Cosine

Cosine, known as "cos," is another primary trigonometric function. It complements sine by representing the ratio of the adjacent side to the hypotenuse in a right triangle. For angle \(\theta\), it is represented as:

• \(\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)

Understanding cosine is vital when solving problems involving right triangles, especially when the length of the adjacent side and hypotenuse are known. For example, if your adjacent side is 4 and the hypotenuse is 5, then \(\cos \theta = \frac{4}{5}\). These ratios are essential for finding unknown angles and sides in geometric problems.

Tangent

The tangent function, abbreviated as "tan," is a unique trigonometric function that is the ratio of the side opposite the angle to the side adjacent to that angle in a right triangle. Expressed as:

• \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\)

Tangent can be particularly useful in real-world applications where forces, inclines, or slopes are involved. If the opposite side is 3 and the adjacent side is 4, then \(\tan \theta = \frac{3}{4}\). It helps to depict the steepness or gradient of a slope, making it critical in both navigation and architecture.

Hypotenuse

The hypotenuse is the longest side of a right triangle and is opposite the right angle. It is a vital reference for calculating trigonometric functions like sine and cosine. To find the hypotenuse, we typically use the Pythagorean theorem, which states that for a right triangle, the square of the hypotenuse \(c\) is equal to the sum of the squares of the other two sides \(a\) and \(b\):

• \(c^2 = a^2 + b^2\)
• \(c = \sqrt{a^2 + b^2}\)

For a point \((4, -3)\), this gives \(c = \sqrt{4^2 + (-3)^2} = \sqrt{25} = 5\). Understanding how to compute the hypotenuse helps solve many problems involving right triangles.

Right Triangle

A right triangle is a type of triangle that includes one angle exactly equal to 90 degrees. This triangle is the basis for understanding trigonometric functions like sine, cosine, and tangent. The sides of a right triangle have unique relationships: the longest side is the hypotenuse, opposite the right angle, while the other two sides are referred to as the "adjacent" and "opposite" sides, depending on the angle in question.
Key features include:

• One angle of 90 degrees
• Pythagorean theorem: \(c^2 = a^2 + b^2\)
• Special angle relationships that define trigonometric functions

These components make the right triangle an essential tool in everything from geometry to engineering, allowing us to solve for missing angles and sides using trigonometry.