Question

For each point, sketch two coterminal angles in standard position whose terminal arm contains the point. Give one positive and one negative angle, in radians, where neither angle exceeds one full rotation. a) (3,5) b) (-2,-1) c) (-3,2) d) (5,-2)

Short answer

a) 7.3136, -5.2528, b) 9.8884, -2.6780, c) 8.8368, -3.7296, d) 5.9027, -6.6637

Step-by-step solution

Understand the Problem

First, determine the initial angle in standard position that reaches the given point \(x, y\). Use the inverse tangent function \(\tan^{-1}(y/x)\) to find the reference angle, then adjust for the correct quadrant.

Sketch Point (3,5)

Find the reference angle for the point (3, 5). The reference angle is \(\theta = \tan^{-1}(5/3)\). In this case, the point is in the first quadrant, so \(\theta = \tan^{-1}(5/3)\). Use a calculator to find \(\theta\).

Identify Coterminal Angles

Coterminal angles are angles that differ from each other by full rotations of \(2\pi\). Find one positive coterminal angle by adding \(2\pi\) to the angle and a negative coterminal angle by subtracting \(2\pi\) from the angle.

Calculate for Point (3, 5)

The initial angle \(\theta \approx 1.0304\) radians. Calculate one positive coterminal angle: \(1.0304 + 2\pi \approx 7.3136\) radians. Calculate a negative coterminal angle: \(1.0304 - 2\pi \approx -5.2528\) radians.

Repeat Steps for Point (-2, -1)

Determine the initial angle using \(\theta = \tan^{-1}(-1/-2) = \tan^{-1}(1/2) \). Since the point is in the third quadrant, add \(\pi\). \(\theta = \pi + \tan^{-1}(1/2) \approx 3.6052\). Calculate one positive (\(3.6052 + 2\pi \ approx 9.8884\)) and one negative (\(3.6052 - 2\pi \ approx -2.6780\)) coterminal angle.

Repeat Steps for Point (-3, 2)

Determine the initial angle using \(\theta = \tan^{-1}(2/-3) \approx -0.5880\). Since the point is in the second quadrant, adjust by adding \(\pi: \theta = \pi - 0.5880 = 2.5536\). Calculate positive (\(2.5536 + 2\pi \approx 8.8368\)) and negative (\(2.5536 - 2\pi \approx -3.7296\)) coterminal angles.

Repeat Steps for Point (5, -2)

Determine the initial angle using \(\theta = \tan^{-1}(-2/5) \approx -0.3805\). Since the point is in the fourth quadrant, \(\theta \approx -0.3805\) is correct already. Calculate positive (\(-0.3805 + 2\pi \approx 5.9027\)) and negative (\(-0.3805 - 2\pi \approx -6.6637)\) coterminal angles.

Key concepts

standard position angles

A standard position angle is an angle whose vertex is at the origin of a coordinate plane and whose initial side lies along the positive x-axis. This is the reference framework for measuring angles in trigonometry and helps us determine the direction and magnitude of the angle. When measuring an angle in standard position, move counterclockwise from the positive x-axis for positive angles and clockwise for negative angles. For example, a 90-degree angle in standard position indicates a quarter turn from the positive x-axis, aligning the terminal side with the positive y-axis. This concept is crucial for understanding angles and their trigonometric properties.

reference angles

A reference angle is the smallest positive angle formed by the terminal side of a given angle and the x-axis. It is always between 0 and 90 degrees (0 and \(\frac{\pi}{2}\)) and helps simplify the calculations of trigonometric functions for larger angles. To find the reference angle, identify in which quadrant the terminal side of the angle is and use the appropriate formula for that quadrant:

• For the first quadrant, the reference angle is the angle itself.
• For the second quadrant, subtract the angle from \(\pi\).
• For the third quadrant, subtract \(\pi\) from the angle.
• For the fourth quadrant, subtract the angle from \(\2 \pi\).

Reference angles are useful in determining the values of trigonometric functions without cumbersome calculations.

radians

A radian is a measure of an angle that describes the relationship between the radius of a circle and the length of an arc. One radian is the angle created when the length of the arc is equal to the radius of the circle. This unit is fundamental in the study of trigonometry and calculus because it naturally connects angles with the properties of circles. One full rotation around a circle is equivalent to \(2 \pi\) radians, which is approximately 6.2832. Unlike degrees, which divide the circle into 360 parts, radians provide a direct measure based on the circle's geometry. Converting between radians and degrees is straightforward: \( \text{\textdegree} \times \left( \frac{\pi}{180} \right) \), while \( \text{radians} \times \left( \frac{180}{\pi} \right) \). This unique measure makes radians an essential tool for advanced mathematical concepts.