Problem 81 - Precalculus

Question

For the following exercises, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by \(t\) lies. $$ \sin (t) > 0 \text { and } \cos (t) > 0 $$

Step-by-Step Solution

Answer

The terminal point determined by \(t\) lies in the first quadrant.

Step 1: Understanding the Problem

We are given that \( \sin(t) > 0 \) and \( \cos(t) > 0 \). We need to find which quadrant of the unit circle these conditions describe. The unit circle is divided into four quadrants, each corresponding to a distinct combination of signs for sine and cosine.

Step 2: Reviewing Quadrant Properties

In the first quadrant, both sine and cosine functions are positive \( (\sin(t) > 0 \text{ and } \cos(t) > 0) \). In the second quadrant, sine is positive and cosine is negative \( (\sin(t) > 0, \cos(t) < 0) \). In the third quadrant, both sine and cosine are negative \( (\sin(t) < 0, \cos(t) < 0) \). In the fourth quadrant, sine is negative and cosine is positive \( (\sin(t) < 0, \cos(t) > 0) \).

Step 3: Identifying the Correct Quadrant

Since we need sine and cosine to both be positive, we look to the properties we reviewed. The only quadrant where both \( \sin(t) \) and \( \cos(t) \) are positive is the first quadrant.

Key Concepts

Quadrant

The coordinate plane, known as the unit circle in trigonometry, is divided into four sections called quadrants. Each of these quadrants has unique characteristics that affect the signs of trigonometric functions.

• First Quadrant: Both sine and cosine functions are positive.
• Second Quadrant: Sine is positive, cosine is negative.
• Third Quadrant: Both sine and cosine are negative.
• Fourth Quadrant: Sine is negative, cosine is positive.

The idea is when you have information about the sign of sine and cosine, you can immediately determine the quadrant. For example, if you know both are positive, you are definitely in the first quadrant. This breakdown is a helpful way to navigate the unit circle.

Sine Function

The sine function is one of the fundamental trigonometric functions associated with the unit circle. It represents the y-coordinate of a point on the circle corresponding to a specific angle. For any angle, the sine value corresponds to how high or low the point is on the unit circle.

• The sine function is positive in the first and second quadrants.
• It is negative in the third and fourth quadrants.
• Maximum value: +1 (when the point is at the top of the circle).
• Minimum value: -1 (when the point is at the bottom of the circle).

A small tip to remember: "Sine is like height," as heights can quickly give you an idea of the sine's positivity or negativity.

Cosine Function

The cosine function, another pillar of trigonometry, represents the x-coordinate of a point on the unit circle. It determines how far left or right a point on the circle is from the origin, based on a given angle.

• The cosine function is positive in the first and fourth quadrants.
• It is negative in the second and third quadrants.
• Maximum value: +1 (when the point is at the extreme right of the circle).
• Minimum value: -1 (when the point is at the extreme left of the circle).

It helps to think of the cosine function as the counterpart to sine, primarily concerned with horizontal positioning rather than vertical.

Trigonometry

Trigonometry is the branch of mathematics that studies the relationships between the angles and lengths of triangles. It is not only vital for understanding geometric concepts but also for analyzing periodic relationships in various fields such as physics and engineering.In trigonometry, the unit circle serves as a powerful tool for defining the main functions — sine, cosine, and tangent. These functions map out a cyclical pattern as the angle moves around the circle, repeating every full rotation (360 degrees or \(2\pi\) radians).

• The unit circle has a radius of 1, which simplifies calculations.
• Angles are typically measured in radians, with one full revolution equivalent to \(2\pi\) radians.
• The unit circle helps visualize how the trigonometric functions translate angles into coordinate values.

Trigonometry reveals the profound periodicity and symmetry inherent in mathematical and physical phenomena, all starting from something as simple as a circle.