Problem 8 - 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 \(\theta .\) $$ (-1,-3) $$

Step-by-Step Solution

Answer

The exact values for the six trigonometric functions of \(\theta\) are: sin\(\theta\) = -\(\sqrt{10}/10\), cos\(\theta\) = -\(\sqrt{10}/10\), tan\(\theta\) = 3, csc\(\theta\) = -\(\sqrt{10}/3\), sec\(\theta\) = -\(\sqrt{10}\), cot\(\theta\) = 1/3

Step 1: Identify the coordinates

First identify the given point, which signifies the coordinates on the terminal side of angle \(\theta\). In this problem, the point is (-1,-3). Since we are in a standard coordinate system, we can identify the x-coordinate as -1 and the y-coordinate as -3.

Step 2: Calculate r

Next, calculate r, which represents the distance from the origin (0,0) to the point (-1,-3). This can be computed using the Pythagorean theorem {r=\(\sqrt{x^2 + y^2}\)}. Substituting the identified x and y coordinates into the formula gives r=\(\sqrt{(-1)^2 + (-3)^2}\)=\(\sqrt{10}\).

Step 3: Calculate Trigonometric Functions

Now that we have x, y, and r, we can calculate the six trigonometric ratios:1. Sin \(\theta\) = y/r = -3/\(\sqrt{10}\) = -\(\sqrt{10}/10\)2. Cos \(\theta\) = x/r = -1/\(\sqrt{10}\) = -\(\sqrt{10}/10\)3. Tan \(\theta\) = y/x = -3/-1 = 34. Csc \(\theta\) = r/y = \(\sqrt{10}\)/-3 = -\(\sqrt{10}/3\)5. Sec \(\theta\) = r/x = \(\sqrt{10}\)/-1 = -\(\sqrt{10}\)6. Cot \(\theta\) = x/y = -1/-3 = 1/3

Key Concepts

coordinate system

The coordinate system is a fundamental mathematical concept used to identify the location of points. Often represented as a grid, a coordinate system allows us to specify a point in space by its position along the horizontal (x-axis) and vertical (y-axis) directions.
The most common type of coordinate system is the Cartesian coordinate system, which divides space into four quadrants. Each quadrant is determined by the signs of the x and y coordinates.

• Quadrant I: x > 0, y > 0
• Quadrant II: x < 0, y > 0
• Quadrant III: x < 0, y < 0
• Quadrant IV: x > 0, y < 0

In our problem, the point (-1,-3) lies in the third quadrant, indicating that both coordinates are negative. This helps us determine the angle's location in the coordinate plane, crucial for finding trigonometric ratios.

Pythagorean theorem

The Pythagorean theorem is a key mathematical relationship used to connect the sides of a right triangle. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
The formula is expressed as: \[ c^2 = a^2 + b^2 \]where:

• \(c\) is the hypotenuse
• \(a\) and \(b\) are the other two sides.

In the context of finding trigonometric functions, the Pythagorean theorem is used to find the distance \( r \) (also known as the hypotenuse) from the origin to the point \((-1, -3)\). We calculate \( r \) as: \[ r = \sqrt{x^2 + y^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]This distance \( r \) is crucial for determining the trigonometric ratios.

trigonometric ratios

Trigonometric ratios are functions that relate the angles and sides of a right triangle. These involve the measurement of angles and are fundamental in geometry. The six trigonometric functions are:

• Sine (\(\sin\theta\))
• Cosine (\(\cos\theta\))
• Tangent (\(\tan\theta\))
• Cosecant (\(\csc\theta\))
• Secant (\(\sec\theta\))
• Cotangent (\(\cot\theta\))

These functions are determined using the three sides of a right triangle: the opposite side, adjacent side, and the hypotenuse.
With respect to our problem located in the coordinate system:

• Sine is \( \sin\theta = \frac{y}{r} \)
• Cosine is \( \cos\theta = \frac{x}{r} \)
• Tangent is \( \tan\theta = \frac{y}{x} \)
• Cosecant is \( \csc\theta = \frac{r}{y} \)
• Secant is \( \sec\theta = \frac{r}{x} \)
• Cotangent is \( \cot\theta = \frac{x}{y} \)

By using our calculated \( r = \sqrt{10} \) along with \( x = -1 \) and \( y = -3 \), we can precisely find these functions' values.

angle in standard position

An angle in standard position is a way of placing an angle in a coordinate plane such that its vertex is at the origin \((0,0)\) and its initial side lies along the positive x-axis.
The orientation and placement in the quadrant are determined by how far the terminal side rotates around the origin, either clockwise (negative angles) or counterclockwise (positive angles).
For our specific problem, since the coordinates \((-1, -3)\) are in the third quadrant, the angle \(\theta\) would be rotated counterclockwise from the positive x-axis, positioning it in the third quadrant. This information is important because it affects the sign and values of the computed trigonometric ratios. In this quadrant, both \(\sin\theta\) and \(\cos\theta\) are negative. Thus understanding the angle's position helps solve trigonometric functions correctly.