Question

Construct a table to indicate the signs of all the trigonometric functions for all four quadrants.

Short answer

Here is the short answer based on the given step-by-step solution: The signs of all six trigonometric functions in each of the four quadrants are as follows: | | I | II | III | IV | |----|---|----|-----|----| |sin | + | + | - | - | |cos | + | - | - | + | |tan | + | - | + | - | |csc | + | + | - | - | |sec | + | - | - | + | |cot | + | - | + | - |

Step-by-step solution

List the quadrants and trigonometric functions in a table format

Start by creating a table with the four quadrants listed as column headers (I, II, III, and IV) and the six trigonometric functions as row headers (sin, cos, tan, csc, sec, cot). | | I | II | III | IV | |----|---|----|-----|----| |sin | | | | | |cos | | | | | |tan | | | | | |csc | | | | | |sec | | | | | |cot | | | | |

Determine the signs of sin, cos, and tan in each quadrant

Recall the properties of the sine, cosine, and tangent functions in relation to the angles in each quadrant. In general: - In quadrant I (0 to \(90^\circ\)), all functions are positive. - In quadrant II (\(90^\circ\) to \(180^\circ\)), sin is positive while cos and tan are negative. - In quadrant III (\(180^\circ\) to \(270^\circ\)), sin and cos are negative, while tan is positive. - In quadrant IV (\(270^\circ\) to \(360^\circ\)), sin is negative, while cos and tan are positive. Update the table accordingly: | | I | II | III | IV | |----|---|----|-----|----| |sin | + | + | - | - | |cos | + | - | - | + | |tan | + | - | + | - |

Determine the signs of csc, sec, and cot in each quadrant

Recall the properties of the reciprocal functions: cosecant (csc), secant (sec), and cotangent (cot). Since they are reciprocals of sine, cosine, and tangent, respectively, their signs will be the same as the signs of their corresponding functions in each quadrant. Update the table accordingly: | | I | II | III | IV | |----|---|----|-----|----| |sin | + | + | - | - | |cos | + | - | - | + | |tan | + | - | + | - | |csc | + | + | - | - | |sec | + | - | - | + | |cot | + | - | + | - | Now we have the completed table, indicating the signs of all six trigonometric functions for each of the four quadrants.

Key concepts

Trigonometric Functions Quadrants

Understanding the behavior of trigonometric functions across different quadrants of the coordinate plane is crucial for students to effectively solve problems in trigonometry. The coordinate plane is divided into four quadrants, based on the signs of the x (horizontal) and y (vertical) coordinates:

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

The angle in a quadrant measures from the positive x-axis counterclockwise. In Quadrant I, the angles range from 0 to 90 degrees. As we move to Quadrants II, III, and IV, the range of angles increases by 90 degrees respectively, completing a full circle at 360 degrees. Each quadrant has its unique influence on the signs of the trigonometric functions: sine (sin), cosine (cos), and tangent (tan). A handy mnemonic to remember this is 'All Students Take Calculus', indicating that in Quadrant I, All functions are positive, in Quadrant II, only the Sine is positive, in Quadrant III, Tan is positive, and in Quadrant IV, Cosine is positive.

Trigonometry Sign Chart

Students can utilize a trigonometry sign chart to quickly determine the sign of a trigonometric function in any given quadrant, without having to plug in specific values or angles. This chart is a visually organized table where the rows represent the trigonometric functions, and the columns represent the four quadrants.

In such a sign chart, positive and negative signs are assigned to each trigonometric function based on their behavior in the specific quadrant. Since reciprocal functions (cosecant, secant, cotangent) share the same sign as their corresponding primary functions (sine, cosine, tangent), they fit neatly into the chart.

For simplification, one can focus initially on sine, cosine, and tangent — since understanding their signs will directly inform the signs of cosecant, secant, and cotangent. Creating and interpreting this chart is essential, as it forms the foundation for solving a variety of trigonometry problems, including evaluating the functions of angles, solving equations, and more.

Sine Cosine Tangent Signs

The signs of sine, cosine, and tangent are vital to grasp as they are fundamental to understanding the behavior of angles and their respective ratios. Each trigonometric function has a specific association with the geometry of a right triangle, which is projected into the circular motion around the coordinate plane.

Sine corresponds to the y-coordinate of the point on the unit circle, cosine to the x-coordinate, and tangent to the ratio of y over x.

Sine (sin)

The sine function takes on positive values in quadrants where the y-coordinate is positive (Quadrants I and II) and negative values where y is negative (Quadrants III and IV).

Cosine (cos)

Similarly, cosine is positive in quadrants where the x-coordinate is positive (Quadrants I and IV) and negative where x is negative (Quadrants II and III).

Tangent (tan)

Tangent, being the ratio of sine to cosine, takes on positive values when both sine and cosine are either both positive or both negative (Quadrants I and III), and negative values when they have opposite signs (Quadrants II and IV).

By understanding these associations, students can confidently identify the signs of these functions for any given angle or quadrant, which is a regular constraint in trigonometric problem solving.