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. \(\sec t, \tan t ; \quad\) quadrant II

Step-by-Step Solution

Answer

\(\sec t = -\sqrt{1 + \tan^2 t}\) in Quadrant II.

Step 1: Understanding the given expression and quadrant

We need to express \(\sec t\) in terms of \(\tan t\) given that the terminal point determined by \(t\) is in quadrant II. In this quadrant, \(\sec t\) is negative and \(\tan t\) is also negative.

Step 2: Using identity for secant and tangent

The trigonometric identity linking secant and tangent is \(\sec^2 t = 1 + \tan^2 t\). We will use this identity to express \(\sec t\) in terms of \(\tan t\).

Step 3: Substitute \(\tan t\) and solve for \(\sec t\)

Start with \(\sec^2 t = 1 + \tan^2 t\). To solve for \(\sec t\), take the square root of both sides to get \(\sec t = \pm \sqrt{1 + \tan^2 t}\).

Step 4: Evaluate signs based on quadrant II

In quadrant II, \(\sec t\) is negative. Therefore, we select the negative root, \(\sec t = -\sqrt{1 + \tan^2 t}\).

Key Concepts

Secant

In trigonometry, the secant function, denoted as \( \sec t \), is the reciprocal of the cosine function. This means \( \sec t = \frac{1}{\cos t} \). It's a crucial identity in trigonometry, often used to simplify complex expressions and solve equations.

Here are some key points about the secant function:

• Reciprocal Relationship: Since \( \sec t \) is the reciprocal of \( \cos t \), it becomes undefined whenever \( \cos t \) equals zero. This occurs at angles of \( 90^\circ \) and \( 270^\circ \), corresponding to \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \) radians.
• Periodic Function: The secant function is periodic with a period of \( 2\pi \), mirroring the cosine function's periodicity.
• Positivity and Negativity: In different quadrants, the sign of the secant function depends on the sign of the cosine. In the second quadrant, \( \sec t \) is negative because \( \cos t \) is negative.

The exercise involves expressing \( \sec t \) in terms of \( \tan t \) while considering quadrant II, where the secant of \( t \) is negative. We apply the identity \( \sec^2 t = 1 + \tan^2 t \) to simplify the expression accordingly.

Tangent

The tangent function, denoted as \( \tan t \), is another fundamental trigonometric function. It is defined as the ratio of the sine function to the cosine function, \( \tan t = \frac{\sin t}{\cos t} \). It illustrates the slope of the angle \( t \) on the unit circle.

Key points to understand about the tangent function include:

• Zero Points: \( \tan t \) is zero whenever \( \sin t = 0 \), which occurs at angles \( 0^\circ, 180^\circ \), and any multiples of \( \pi \) radians, because sine is zero at these points.
• Undefined Points: Tangent is undefined and has vertical asymptotes whenever \( \cos t = 0 \), at angles \( 90^\circ, 270^\circ \), and at every multiple of odd \( \frac{\pi}{2} \) radians, due to division by zero.
• Sign in Quadrant II: In the second quadrant, \( \tan t\) is negative because the sine is positive and cosine is negative. This property affects how one would select the correct sign for expressions involving tangent.

When an exercise requires connecting \( \sec t \) to \( \tan t \), understanding these core properties and their behaviors in different quadrants is essential for accurate translations between trigonometric functions.

Quadrant II

The second quadrant is one of the four quadrants in the Cartesian coordinate system where angles measure between \( 90^\circ \) (\( \frac{\pi}{2} \) radians) and \( 180^\circ \) (\( \pi \) radians). In this quadrant, trigonometric functions make distinct transitions in terms of their signs.

Important traits of Quadrant II include:

• Angle Ranges: Quadrant II covers angles beyond a right angle and less than half a full circle, specifically between \( 90^\circ \) and \( 180^\circ \).
• Sign of Functions: Here, sine is positive, while cosine and tangent functions are negative. This inversion is influenced by the position on the unit circle, where the x-values (cosine) become negative and y-values (sine) remain positive.
• Real-world Context: This quadrant might simulate scenarios where the direction is west of north or east of south, useful in navigation tasks.
• Application in Exercises: When solving trigonometric problems involving quadrant II, one must carefully consider these sign changes to find the correct angle or function values.

In the context of the exercise, knowing that both \( \sec t \) and \( \tan t \) are negative in quadrant II aids in correctly resolving their expressions and understanding the calculated signs.