Problem 29 - Algebra and Trigonometry

Question

find the exact value of each of the remaining trigonometric functions of \(\theta\) $$ \tan \theta=-\frac {2}{3}, \quad \sin \theta>0 $$

Step-by-Step Solution

Answer

The exact values of the remaining trigonometric functions of \(\theta\): \(\cos\theta = -\sqrt{5}/3\), \(\sin\theta = 2\sqrt{5}/5\), \(\cot\theta = 3/2\), \(\sec\theta = -3\sqrt{5}/5\), \(\csc\theta = \sqrt{5}/2\).

Step 1: Determine quadrant

The sign of the sine function indicates that the angle \(\theta\) is in the second or first quadrant, but since the tangent of this angle is negative, it could only be in the second quadrant where \(\sin \theta > 0\) and \(\tan \theta < 0\). So, \(\theta\) is in the second quadrant.

Step 2: Using Pythagorean Identity

Utilize the Pythagorean Identity \(1 + \tan^2\theta = \sec^2\theta\) or \(1 = \cos^2\theta + \sin^2\theta\) to find the cosine. Replace \(\tan \theta\) by \(-2/3\) in the identity. So, \(\cos \theta = \sqrt{1 - \sin^2\theta}\). The value of \(\sin \theta\) can be found as \(\sin \theta = \sqrt{1 - \cos^2 \theta}\). Keep in mind that the cosine function in the second quadrant is negative.

Step 3: Determine the value of sine and cosine

Apply the definitions, \(\cos\theta= \sqrt{1 - \tan^2\theta}\) and \(\sin\theta= \sqrt{1 - \cos^2\theta}\), then \(\cos\theta = - \sqrt{1 - (-2/3)^2} = -\sqrt{5}/3\) and \(\sin\theta = \sqrt{1 - (-\sqrt{5}/3)^2} = 2\sqrt{5}/5\)

Step 4: Finding all trigonometric function values

\(\cot\theta = -1/\tan\theta = 3/2\), \(\sec\theta = 1/\cos\theta = -3/\sqrt{5} = -3\sqrt{5}/5\), \(\csc\theta= 1/\sin\theta = 5/2\sqrt{5} = 5\sqrt{5}/10 = \sqrt{5}/2\)

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that comes from the Pythagorean theorem. It expresses the relationship between the sine, cosine, and tangent of an angle. In its simplest form, it is written as: \[1 = \sin^2\theta + \cos^2\theta\] This identity shows that for any angle \(\theta\), the square of the sine function plus the square of the cosine function is always equal to one. This is immensely useful in solving trigonometric problems because, with two of these functions given, you can always find the third. Use the identity to derive other forms like:

• \(1 + \tan^2\theta = \sec^2\theta\)
• \(1 + \cot^2\theta = \csc^2\theta\)

In our exercise, when given \(\tan \theta = -\frac{2}{3}\), we can use the form \(1 + \tan^2\theta = \sec^2\theta\) to find \(\sec \theta\), and then determine both \(\cos \theta\) and \(\sin \theta\) using this identity, ensuring all values conform to the given quadrant conditions.

Second Quadrant Angles

Understanding the quadrant system in the unit circle is crucial for solving trigonometric problems. The second quadrant, where our angle \(\theta\) resides in this exercise, spans angles from \(90^\circ\) to \(180^\circ\). Here, the sine function is positive, but the cosine and tangent functions are negative. For angles in this quadrant:

• \(\sin \theta > 0\)
• \(\cos \theta < 0\)
• \(\tan \theta < 0\)

These sign rules help us check and validate the solutions we derive in our exercise. Since \(\sin \theta > 0\) and \(\tan \theta < 0\), it confirms \(\theta\) exists in this quadrant, ensuring the calculated values for \(\cos \theta\) and other functions are correct relative to their signs.

Tangent Function

The tangent function, \(\tan \theta\), is a primary trigonometric function related to the lengths of the opposite and adjacent sides of a right triangle. It can also be expressed as the ratio of sine to cosine: \[\tan \theta = \frac{\sin \theta}{\cos \theta}\] In this particular problem, you are given \(\tan \theta = -\frac{2}{3}\). This initially tells us that for an angle \(\theta\), the opposite side is 2 units and the adjacent side is 3 units in a right triangle setup. The negative sign indicates that \(\theta\) lies in a quadrant where tangent values are negative (second or fourth quadrants). In our exercise, it is verified that \(\theta\) is in the second quadrant, conforming with \(\tan \theta < 0\).
Additionally, knowing the value of tangent helps find the values of other trigonometric functions using the identities:

• \(\cot \theta = \frac{1}{\tan \theta}\)
• Using the Pythagorean identity form : \(1 + \tan^2\theta = \sec^2\theta\)

This information becomes instrumental in solving and verifying other trigonometric identities and functions related to \(\theta\).