Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(-8,15)$$

Step-by-Step Solution

Answer

The six trigonometric functions are: \(\sin\theta = \frac{15}{17}\), \(\cos\theta = -\frac{8}{17}\), \(\tan\theta = -\frac{15}{8}\), \(\csc\theta = \frac{17}{15}\), \(\sec\theta = -\frac{17}{8}\), \(\cot\theta = -\frac{8}{15}\).

Step 1: Identify the Quadrant

The given point is \((-8, 15)\). Because the x-coordinate is negative and the y-coordinate is positive, the point lies in the second quadrant.

Step 2: Calculate the Reference Angle

The reference angle can be found using the coordinates of the point. The terminal point \((-8, 15)\) represents the opposite and adjacent sides of a right triangle. Use the tangent function: \( \tan \theta = \frac{15}{-8} \). The reference angle is \( \theta = \arctan\left(\frac{15}{8}\right) \).

Step 3: Find the Hypotenuse using the Pythagorean Theorem

To find the hypotenuse (r), use the formula: \( r = \sqrt{x^2 + y^2} \). Substitute for \(x\) and \(y\): \( r = \sqrt{(-8)^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \).

Step 4: Calculate Trigonometric Functions

Using the point \((-8, 15, 17)\): - \( \sin\theta = \frac{15}{17} \)- \( \cos\theta = \frac{-8}{17} \)- \( \tan\theta = \frac{15}{-8} = -\frac{15}{8} \)- \( \csc\theta = \frac{17}{15} \)- \( \sec\theta = \frac{17}{-8} = -\frac{17}{8} \)- \( \cot\theta = \frac{-8}{15} \)

Step 5: Determine the Angle in Standard Position

The calculated trigonometric functions describe an angle \(\theta\) in the second quadrant. Since we know the point \((-8, 15, 17)\) defines the terminal side in the second quadrant, the angle \(\theta\) is the supplement of the reference angle. However, the trigonometric functions already reflect \(\theta\) in its correct quadrant placement.

Key Concepts

Standard Position

A standard position of an angle is a way to set up an angle in a coordinate system. An angle is said to be in standard position if its vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis. The terminal side of the angle rotates counterclockwise from the initial side for a positive angle.
When an angle is described in this manner, it helps us understand and calculate the trigonometric functions. All points on the terminal side of the angle describe a relationship with sine, cosine, and tangent based on their distances from the x and y-axis. Together, they help determine the angle's properties in standard position.

• The initial side is fixed on the x-axis.
• The angle's measure increases as it moves counterclockwise.
• Important for analyzing angles and determining quadrant placements.

Reference Angle

The reference angle is a helpful concept when dealing with angles on the coordinate plane. It is the acute angle formed by the terminal side of the given angle and the x-axis.
Reference angles are always between 0° and 90°, making them tools for simplifying calculations of trigonometric functions. Since the trigonometric values in different quadrants can be related back to the reference angle, they help in determining the correct trigonometric function values.
To find a reference angle:

• Measure the distance between the terminal side of the angle and the nearest x-axis.
• For angles in the second quadrant, the reference angle is calculated as \(180^\circ - \theta\).
• It offers simplicity and eases complexity when working with non-standard angles.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle used to relate the sides of a right triangle. Formulated as \(a^2 + b^2 = c^2\), the theorem helps calculate the hypotenuse when the other two sides are known.
In the context of trigonometry, it allows us to determine the length of the hypotenuse (r) when the terminal side of an angle in standard position passes through a point—such as (-8, 15).

• Calculates hypotenuse: \( r = \sqrt{x^2 + y^2} \).
• Helps determine the distance from the origin to the point on the terminal side.
• Key for ensuring accurate trigonometric computations when using coordinates.

Second Quadrant

In a coordinate plane divided by four quadrants, the second quadrant is located at the top-left area. Here, the x-values are negative, while the y-values are positive.
This quadrant is significant when determining the signs of trigonometric functions:

• In the second quadrant, sine functions are positive because they represent the y-coordinate.
• Cosine and tangent functions are negative, arising from negative x-values.
• Understanding the quadrant helps apply the correct sign to trigonometric values. It also aids in identifying reference angles and positioning angles correctly in standard position.