Problem 57 - Algebra and Trigonometry

Question

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\tan t, \sin t ; \quad\) Quadrant IV

Step-by-Step Solution

Answer

\( \tan t = \frac{\sin t}{\sqrt{1-\sin^2 t}} \) in Quadrant IV.

Step 1: Recall Trigonometric Identities

In any quadrant, the basic identity connecting tangent and sine is \( \tan t = \frac{\sin t}{\cos t} \). We need to express \( \tan t \) in terms of \( \sin t \).

Step 2: Determine Signs in Quadrant IV

In Quadrant IV, \( \sin t \) is negative and \( \cos t \) is positive. This affects the signs of the trigonometric functions that we need to consider when expressing \( \tan t \).

Step 3: Substitute Relationships

From the Pythagorean identity, \( \cos^2 t = 1 - \sin^2 t \). Since \( \cos t \) is positive in Quadrant IV, we find \( \cos t = \sqrt{1-\sin^2 t} \).

Step 4: Express \( \tan t \) in Terms of \( \sin t \)

Substitute the expression for \( \cos t \) from Step 3 into the identity from Step 1: \( \tan t = \frac{\sin t}{\sqrt{1-\sin^2 t}} \). Since both \( \tan t \) and \( \sin t \) are consistent with their signs in Quadrant IV, this is the final expression for \( \tan t \) in terms of \( \sin t \).

Key Concepts

Pythagorean identity

The Pythagorean identity is a cornerstone in trigonometry. It relates the square of the sine function and the square of the cosine function. The most familiar form is: \[\sin^2 t + \cos^2 t = 1\] This equation essentially states that for any angle \( t \), the sum of the squares of the sine and cosine of that angle is always equal to one. This identity is immensely useful because it allows us to express one trigonometric function in terms of the other.
For instance, if you know \( \sin t \), you can determine \( \cos t \) by rearranging the identity to find: \[\cos^2 t = 1 - \sin^2 t\] By taking the square root, you can find \( \cos t \) and use this relationship in various calculations. Understanding this identity simplifies working with more complex expressions and solving equations involving trigonometric functions.

tangent and sine relationship

Trigonometric functions are interrelated, and one common relationship is between tangent and sine. The basic relationship is given by the equation: \[\tan t = \frac{\sin t}{\cos t}\] This formula tells us that the tangent of an angle \( t \) is the ratio of its sine to its cosine. This relationship is particularly beneficial when you're asked to express tangent in terms of sine, as it involves manipulating the formula to focus on \( \sin t \).
It is important to keep in mind the signs of these functions based on the quadrant they reside in, as this affects their values and relationships. In Quadrant IV, where the cosine is positive and sine is negative, this directly impacts the value of tangent. The formula becomes pivotal when the sine or cosine is directly known, making it easier to switch between the functions.

quadrants in trigonometry

Trigonometry divides the unit circle into four quadrants, each with distinct characteristics that affect the sign of trigonometric functions. Here's a quick guide:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, cosine is positive.

In Quadrant IV, the unique combination of signs means that while sine is negative, cosine remains positive. This directly affects other functions like tangent, where the sign follows the division \( \frac{\sin t}{\cos t} \), resulting sometimes in negative values. Understanding the characteristics of each quadrant helps in predicting the behavior of trigonometric functions in various calculations, ensuring accurate solution of trigonometric equations.