Problem 115 - A Graphical Approach to Precalculus with Limits

Question

Find all trigonometric function values for each angle \(\boldsymbol{\theta}\). \(\tan \theta=-\frac{15}{8},\) given that \(\theta\) is in quadrant II

Step-by-Step Solution

Answer

\(\sin \theta = \frac{15}{17}\), \(\cos \theta = -\frac{8}{17}\), \(\tan \theta = -\frac{15}{8}\).

Step 1: Understanding the Problem

We are given \(\tan \theta = -\frac{15}{8}\) and that \(\theta\) is in quadrant II. We need to find sine, cosine, and tangent values.

Step 2: Calculate Hypotenuse Using Pythagorean Theorem

Knowing the adjacent side is 8 and the opposite side is -15 (since tangent = opposite/adjacent), we apply the Pythagorean theorem. The hypotenuse \(r\) is computed as \(r = \sqrt{(-15)^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17\).

Step 3: Understanding Sign Convention in Quadrant II

In quadrant II, sine values are positive and cosine values are negative. We will use this information to determine the signs of the other trigonometric values.

Step 4: Calculate Sine (sin) Value

Using the triangle sides, \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{-15}{17}\). Since sine is positive in quadrant II, \(\sin \theta = \frac{15}{17}\).

Step 5: Calculate Cosine (cos) Value

Using the triangle sides, \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{8}{17}\). Since cosine is negative in quadrant II, \(\cos \theta = -\frac{8}{17}\).

Step 6: Verify Tangent (tan) Value

We verify \(\tan \theta\) using the calculated sine and cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{15}{17}}{-\frac{8}{17}} = -\frac{15}{8}\). This matches the given condition.

Key Concepts

Quadrant Analysis

Quadrant analysis helps us determine the sign of trigonometric functions based on the angle's position. The coordinate plane is divided into four quadrants. Quadrant I has all positive trigonometric values, whereas in Quadrant II, sine values are positive, but cosine and tangent are negative.
This is because when you are in Quadrant II, the angle \( \theta \) corresponds to an x-value (adjacent side) that is negative and a y-value (opposite side) that is positive, reflecting the Cartesian coordinate system.
Understanding quadrant analysis lets us predict the sign of trigonometric functions without calculating the actual values. This knowledge is particularly helpful when solving trigonometry problems, as tracing back from the tangent ratio in Quadrant II, shows us that sine will be positive, while cosine will be negative. This aids in verifying whether we have located the correct angle or value.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry used to relate the sides of a right-angled triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it is written as: \( c^2 = a^2 + b^2 \).
In trigonometry, especially when dealing with trigonometric functions, the theorem helps in defining relationships between sine, cosine, and tangent. Knowing two sides of a right triangle, we can always find the third using this theorem.
For instance, if the opposite side is -15 and the adjacent side is 8 in our problem, we can compute the hypotenuse as: \( r = \sqrt{(-15)^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17 \).
With the hypotenuse known, we can simplify and find the sine and cosine values of the angle by dividing the lengths of the opposite and adjacent sides by the hypotenuse.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the involved variables. They provide essential relationships that simplify the understanding of trigonometric functions.
Some fundamental identities include the \( \sin^2 \theta + \cos^2 \theta = 1\), which follows from the Pythagorean theorem, and helps in finding one trigonometric function if the other is known. Another useful identity is the tangent identity, \( \ an \theta = \rac{\sin \theta}{\cos \theta}\), which helps us verify values against given conditions.
Understanding these identities simplifies solving complex trigonometric problems. In the problem we have, using \( \ an \theta = \rac{\sin \theta}{\cos \theta}\) ensures that our calculation matches the given ratio of \( \ an \theta = -\frac{15}{8}\). Knowing and applying these identities ensure correct solutions and can make trigonometry much easier to understand and apply.
Explore and practice these identities to gain a deeper insight into the importance and utility of trigonometric functions in different mathematical and real-world applications.