Problem 26 - Precalculus: A Concise Course

Question

Find the values of the six trigonometric functions of \(\boldsymbol{\theta}\) with the given constraint. $$ \cos \theta=-\frac{4}{5} \quad \theta \text { lies in Quadrant III } $$

Step-by-Step Solution

Answer

The values of the six trigonometric functions are: \(\sin(\theta) = -\frac{3}{5}\), \(\cos(\theta) = -\frac{4}{5}\), \(\tan(\theta) = \frac{3}{4}\), \(\csc(\theta) = -\frac{5}{3}\), \(\sec(\theta) = -\frac{5}{4}\), and \(\cot(\theta) = \frac{4}{3}\).

Step 1: Determine sine's sign

To know the value of sine in the third quadrant, we must understand the ASTC rule or All Students Take Calculus rule which states that in an x-y plane, All functions are positive in the first quadrant, Sin is positive in the second, Tangent in the third and Cosine in the fourth quadrant. Since \(\theta\) lies in the third quadrant, \(\sin(\theta)\) must be negative.

Step 2: Calculate sine value

We can determine the sine value using the Pythagorean identity which states that \(\sin^2(\theta) + \cos^2(\theta) = 1\). We have the value of cosine as \(-4/5\). Substituting the value of cosine into the equation we have \(\sin^2(\theta) = 1 - (-\frac{4}{5})^2 = \frac{9}{25}\). So, \(\sin(\theta) = \pm\sqrt{\frac{9}{25}} = \pm\frac{3}{5}\). Since we determined in step 1 that \(\sin(\theta)\) is negative in the third quadrant, we can say \(\sin(\theta) = -\frac{3}{5}\).

Step 3: Calculating other trigonometric function values

Now we can calculate the other functions using the relationships \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Substituting the given and calculated values, we obtain the function values: \(\tan \theta = \frac{-3/5}{-4/5} = \frac{3}{4}\), \(\csc \theta = \frac{1}{-3/5} = -\frac{5}{3}\), \(\sec \theta = \frac{1}{-4/5} = -\frac{5}{4}\), \(\cot \theta = \frac{1}{3/4} = \frac{4}{3}\).

Step 4: Compiling all the answers

We have already found all the required trigonometric function values. So, we record all our findings.

Key Concepts

ASTC rule

The ASTC rule is a simple mnemonic to remember the signs of trigonometric functions in different quadrants of the coordinate plane. 'ASTC' stands for 'All Students Take Calculus', indicating that

• All - All trigonometric functions are positive in Quadrant I
• Students - Only Sine and its reciprocal function, cosecant, are positive in Quadrant II
• Take - Only Tangent and its reciprocal function, cotangent, are positive in Quadrant III
• Calculus - Only Cosine and its reciprocal function, secant, are positive in Quadrant IV

In the context of the given exercise, we understand that since the angle \(\theta\) lies in Quadrant III, the cosine value is negative, and accordingly, the sine will also be negative as dictated by the ASTC rule.

Pythagorean identity

The Pythagorean identity is a fundamental relation in trigonometry expressing that for any angle \(\theta\), the squares of the sine and cosine of \(\theta\) add up to 1: \[\sin^2(\theta) + \cos^2(\theta) = 1.\] This equation is based on the Pythagorean theorem related to the sides of a right triangle. To apply the Pythagorean identity to calculate the sine value in the third quadrant,
one must substitute the known cosine value and solve for the sine function. Because \(\cos\theta = -\frac{4}{5}\), the Pythagorean identity allows us to find that \(\sin^2\theta = 1 - \left(-\frac{4}{5}\right)^2 = \frac{9}{25}\), leading to a sine value of \(\pm\frac{3}{5}\). Since we're in Quadrant III, we choose the negative value for sine.

Sine and cosine values

The sine and cosine values for a specific angle provide the ratio of the sides of a right-angled triangle to the length of the hypotenuse. Specifically, the cosine of an angle represents the adjacent side over the hypotenuse, while the sine represents the opposite side over the hypotenuse. In the context of our problem, since \(\cos\theta = -\frac{4}{5}\), we are looking at a triangle where the length of the adjacent side is 4 units and the hypotenuse is 5 units, both considered with their respective signs. The sine's absolute value is determined using the Pythagorean identity, and the negative sign is affixed because the opposite side in Quadrant III is in the negative y-direction.

Calculating trigonometric functions

Calculating trigonometric functions involves finding the values for the six main functions: sine, cosine, tangent, cosecant, secant, and cotangent. When given one function, one can often find the others through identities and relationships between them. For example, tangent is the ratio of sine to cosine, and the reciprocal functions (cosecant, secant, and cotangent) are simply the inverses of sine, cosine, and tangent, respectively. In the solved exercise, the tangent was found by dividing sine by cosine, and similarly, the other functions were found using their definitions based on the given cosine value and the calculated sine value.