7.4 Computing Trigonometric Functions

Signs of Trigonometric Functions in Different Quadrants

• Understanding the signs of trigonometric functions across the four quadrants is fundamental in analyzing and computing these functions for any given angle.

• In Quadrant I, where both x and y are positive, all six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are positive.

• In Quadrant II, x is negative and y is positive. Here, sine and cosecant are positive, while cosine, secant, tangent, and cotangent are negative.

• In Quadrant III, both x and y are negative. The tangent and cotangent functions are positive in this quadrant, as they are ratios of y/x and x/y, respectively, both of which are positive when x and y are negative. The other four functions are negative.

• In Quadrant IV, x is positive while y is negative. Cosine and secant are positive, whereas sine, cosecant, tangent, and cotangent are negative.

Summary of signs:

• Quadrant I: All six functions are positive.

• Quadrant II: Sine and cosecant are positive; others negative.

• Quadrant III: Tangent and cotangent are positive; others negative.

• Quadrant IV: Cosine and secant are positive; others negative.

Coterminal Angles

Two angles in standard position are said to be coterminal if they share the same terminal side when drawn in the coordinate plane. This means:

• They differ by a full rotation of 360° (2π radians).

• Their trigonometric function values are equal because they have the same terminal side.

Mathematically:

• For degrees: $\theta_{\cot er\min al}$ ​= θ+360∘×k where k∈Z

• For radians: $\theta_{\cot er\min al}$ = θ+2π×k where k∈Z

Example: Given the angle theta equals π divided by 6, the following calculations show examples of a positive and a negative coterminal angle.

• A positive coterminal angle is found by adding 2π:

• (π / 6) + 2π = (13π / 6)

• A negative coterminal angle is found by subtracting 2π:

• (π / 6) - 2π = (-11π / 6)

Finding Coterminal Angles

To find coterminal angles:

• Add or subtract multiples of 360° (2π radians).

• This process helps locate angles that share the same terminal side but differ by full rotations.

Practical steps:

1. Take the given angle.

2. Add 360° or 2π radians repeatedly to find positive coterminal angles.

3. Subtract 360° or 2π radians repeatedly to find negative coterminal angles.

This is useful in simplifying trigonometric calculations and understanding periodicity.

Reference Angles

A reference angle is the acute angle formed between the terminal side of a given angle ($\theta$) and the x-axis. It is always between 0° and 90° (or 0 and (π/2) radians).

Purpose of reference angles:

• Simplify the calculation of trigonometric functions for angles outside the first quadrant.

• Allow the use of known values of sine, cosine, and tangent from the first quadrant.

How to find the reference angle:

• For angles in Quadrant I, the reference angle is simply ($\theta$).

• For angles in Quadrant II, the reference angle is ($\theta$' = 180° - $\theta$) or (π - $\theta$).

• For angles in Quadrant III, it is ($\theta$' = $\theta$ - 180°) or ($\theta$ - π).

• For angles in Quadrant IV, it is ($\theta$' = 360° - $\theta$) or (2π - $\theta$).

Example: For ($\theta$ = 135°), which lies in Quadrant II, the reference angle is: [ 180° - 135° = 45° ]

Using Reference Angles to Calculate Trigonometric Functions

Once the reference angle ($\theta$') is known, the values of trigonometric functions for the original angle ($\theta$) can be derived by:

• Using the positive value of the function from the reference angle.

• Applying the sign based on the quadrant in which ($\theta$) lies.

Procedure:

1. Find the reference angle ($\theta$').

2. Determine the sign of the function based on the quadrant.

3. Use known values of the functions at ($\theta$') (often from the unit circle).

4. Assign the correct sign to obtain the value for ($\theta$).

Example: Find (\cos 135°):

• Reference angle: (180° - 135° = 45°).

• (\cos 45° = \frac{\sqrt{2}}{2}).

• Since (135°) is in Quadrant II, where cosine is negative, [ \cos 135° = -\frac{\sqrt{2}}{2} ]

Examples of Calculating Trigonometric Functions from Given Angles and Conditions

Example 1: Find coterminal angles

Given (\theta = \frac{\pi}{6}):

• Positive coterminal angle:[ \frac{\pi}{6} + 2\pi = \frac{\pi}{6} + \frac{12\pi}{6} = \frac{13\pi}{6} ]

• Negative coterminal angle:[ \frac{\pi}{6} - 2\pi = \frac{\pi}{6} - \frac{12\pi}{6} = -\frac{11\pi}{6} ]

Example 2: Find the reference angle and compute (\cos \theta) for (\theta = 135^\circ)

• Reference angle:[ 180^\circ - 135^\circ = 45^\circ ]

• (\cos 135^\circ):[ \cos 45^\circ = \frac{\sqrt{2}}{2} ] Since (135^\circ) is in Quadrant II, [ \cos 135^\circ = -\frac{\sqrt{2}}{2} ]

Example 3: Find all remaining trigonometric functions for (\sin \theta = -\frac{3}{5}), with (\pi < \theta < \frac{3\pi}{2})

• The given (\sin \theta) is negative, and (\theta) is in Quadrant III (since (\pi < \theta < 3\pi/2)).

• Step 1: Find (\cos \theta):

  Using the Pythagorean identity: [ \sin^2 \theta + \cos^2 \theta = 1 ] [ \left(-\frac{3}{5}\right)^2 + \cos^2 \theta = 1 ] [ \frac{9}{25} + \cos^2 \theta = 1 ] [ \cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25} ] [ \cos \theta = \pm \frac{4}{5} ]

• Step 2: Determine the sign of (\cos \theta):

  Since (\theta) is in Quadrant III, both (\sin \theta) and (\cos \theta) are negative:

  [ \cos \theta = -\frac{4}{5} ]

• Step 3: Find other functions:

  • Tangent:[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{3}{5}}{-\frac{4}{5}} = \frac{-3/5}{-4/5} = \frac{-3}{-4} = \frac{3}{4} ]

  • Cosecant:[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3} ]

  • Secant:[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4} ]

  • Cotangent:[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{3}{4}} = \frac{4}{3} ]

This comprehensive overview covers the fundamental properties and methods for working with trigonometric functions, including their signs in different quadrants, coterminal angles, reference angles, and practical calculation examples. Mastery of these concepts is essential for solving a wide variety of trigonometric problems efficiently and accurately.