Problem 2 - 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 \(\theta .\) $$ (-12,5) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the sides of right triangle

First, the point (-12,5) forms a right triangle with the origin. This is because the x-coordinate corresponds to the adjacent side (-12) and the y-coordinate to the opposite side (5). In this notation, negative values mean that the distance is to the left of the origin for the x-coordinate or below the origin for the y-coordinate.

Step 2: Calculate the Hypotenuse

Using the Pythagorean theorem, \(c = \sqrt{a^2 + b^2} \), calculate the length of the hypotenuse. In our case, the 'a' corresponds to the x-coordinate which is -12, and 'b' corresponds to the y-coordinate which is 5. Hence, the hypotenuse is \(c = \sqrt{(-12)^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13\).

Step 3: Calculate Trigonometric Values

The six trigonometric functions of the angle can now be calculated as follows:- Sine (\(sin(\theta)\)) is \( \frac{Opposite}{Hypotenuse} = \frac{5}{13} \),- Cosine (\(cos(\theta)\)) is \( \frac{Adjacent}{Hypotenuse} = \frac{-12}{13} \) because the x-axis value is to the left,- Tangent (\(tan(\theta)\)) is \( \frac{Opposite}{Adjacent} = -\frac{5}{12} \),- Cosecant (\(csc(\theta)\)) is \( \frac{1}{sin(\theta)} = \frac{13}{5} \),- Secant (\(sec(\theta)\)) is \( \frac{1}{cos(\theta)} = -\frac{5}{12} \),- Cotangent (\(cot(\theta)\)) is \( \frac{1}{tan(\theta)} = -\frac{12}{5} \).Note: The negative sign value indicates in which quadrant the angle lies.

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This distinct property makes it a fundamental piece in trigonometry. When working with points in the coordinate plane, such as the point \((-12,5)\), you can imagine this point forming a right triangle with the origin (0,0).
Here, the x-coordinate \(-12\) serves as the base (adjacent side) along the x-axis and the y-coordinate \(5\) is the height (opposite side) along the y-axis. These two sides meet at a right angle on the coordinate grid.
Why is this important? The relationship between the sides of a right triangle is what allows trigonometric functions to be defined and used for various calculations.

Pythagorean Theorem

The Pythagorean Theorem is a crucial concept in geometry and trigonometry. It describes the relationship between the sides of a right triangle. Mathematically, it is expressed as \(c = \sqrt{a^2 + b^2}\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.
This theorem allows us to calculate the hypotenuse when the other two sides are known. For our example with sides \(-12\) and \(5\), the hypotenuse is determined to be \(13\) using this formula.
The Pythagorean Theorem not only helps with these calculations but is also invaluable in proving concepts related to distance and establishing properties of trigonometric ratios.

Sine Cosine Tangent

Sine, cosine, and tangent are the primary trigonometric functions that relate the angles of a right triangle to its side lengths. These functions are defined as:

• Sine \(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
• Cosine \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
• Tangent \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)

Using the triangle formed by the point \((-12,5)\), we calculate:

• \(\sin(\theta) = \frac{5}{13}\)
• \(\cos(\theta) = \frac{-12}{13}\)
• \(\tan(\theta) = -\frac{5}{12}\)

These functions are central to solving problems involving angles and distances.

Cosecant Secant Cotangent

The secondary trigonometric functions are cosecant, secant, and cotangent, which are the reciprocals of sine, cosine, and tangent:

• Cosecant \(\csc(\theta) = \frac{1}{\sin(\theta)}\)
• Secant \(\sec(\theta) = \frac{1}{\cos(\theta)}\)
• Cotangent \(\cot(\theta) = \frac{1}{\tan(\theta)}\)

For the angle \(\theta\) with point \((-12,5)\), these values are:

• \(\csc(\theta) = \frac{13}{5}\)
• \(\sec(\theta) = -\frac{13}{12}\)
• \(\cot(\theta) = -\frac{12}{5}\)

These functions provide further insights and applications when exploring relationships within right triangles.