Problem 62 - 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. \(\sin t, \sec t ; \quad\) Quadrant \(\mathrm{IV}\)

Step-by-Step Solution

Answer

\(\sin t = -\sqrt{1 - \frac{1}{\sec^2 t}}\)

Step 1: Understand the Quadrant IV Trigonometric Signs

In Quadrant IV, the sine function (\(\sin t\)) is negative and the secant function (\(\sec t = \frac{1}{\cos t}\)) is positive, because cosine is positive in Quadrant IV. This step helps us understand the signs of the trigonometric functions involved.

Step 2: Relate Sine and Cosine

We know that \(\sin^2 t + \cos^2 t = 1\) for all values of \(t\). Since \(\sec t = \frac{1}{\cos t}\), we can re-write \(\cos t = \frac{1}{\sec t}\). So substitute this into \(\sin^2 t + \cos^2 t = 1\) to express \(\sin t\):\[\sin^2 t + \left(\frac{1}{\sec t}\right)^2 = 1\].

Step 3: Solve for \(\sin t\) in Terms of \(\sec t\)

From the equation \(\sin^2 t + \frac{1}{\sec^2 t} = 1\), isolate \(\sin^2 t\):\[\sin^2 t = 1 - \frac{1}{\sec^2 t}\].Now, find \(\sin t\) knowing it is negative in Quadrant IV: \(\sin t = -\sqrt{1 - \frac{1}{\sec^2 t}}\).

Key Concepts

Sine Function in Quadrant IV

The sine function in Quadrant IV has an interesting behavior. The angle or terminal side in Quadrant IV corresponds to a value where the sine is negative. This is due to the vertical orientation of the angle under consideration. Here are some key points:

• In Quadrant IV, the y-coordinates, which represent the sine values, are below the x-axis, rendering them negative.
• This negativity follows from the definition of the sine function, which is the ratio of the opposite side to the hypotenuse in a right triangle.

The significance of sine being negative in this quadrant is vital when solving trigonometric identities and equations, as it directly influences the sign of the result when finding other trigonometric functions derived from sine.

Secant Function

When solving trigonometric problems, the secant function frequently comes into play. The secant, represented as sec, is defined as the reciprocal of the cosine function, i.e., \(\sec t = \frac{1}{\cos t}\). Here’s why the secant function is essential in trigonometric identities:

• Since secant is the reciprocal of cosine, and cosine is positive in Quadrant IV, this makes the secant function also positive there.
• Understanding secant helps in transforming and simplifying complex trigonometric expressions, specifically when cosine values are small, and secant values become large.

Knowing that secant is positive above the x-axis influences how you approach problems involving other trigonometric ratios in different quadrants.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometric concepts. It states that for any angle \(t\), the expression \(\sin^2 t + \cos^2 t = 1\) holds. This identity is crucial for several reasons:

• It provides a link between the sine and cosine functions, showing how they maintain a harmonious relation through the unit circle.
• In simplified forms like \(\cos t = \frac{1}{\sec t}\), you can model complex trigonometric expressions involving secant and sine, as seen in the process of solving the original exercise.

By understanding the Pythagorean identity, we deduce relationships between different functions, unlocking new ways to express and solve trigonometric equations, especially when dealing with unknown sides or angles in triangles.