Problem 97 - Precalculus

Question

Determine whether each statement is true or false. It is possible for all six trigonometric functions of the same angle to have positive values.

Step-by-Step Solution

Answer

True, all six functions can be positive in the first quadrant.

Step 1: Identify the Six Trigonometric Functions

The six trigonometric functions are sine (\( ext{sin} \)), cosine (\( ext{cos} \)), tangent (\( ext{tan} \)), cosecant (\( ext{csc} \)), secant (\( ext{sec} \)), and cotangent (\( ext{cot} \)). We need to determine if all of these can be positive simultaneously.

Step 2: Understand Positivity in Quadrants

In the coordinate plane, angles belong to one of four quadrants. Sine and cosecant are positive in the first and second quadrants; cosine and secant are positive in the first and fourth quadrants; tangent and cotangent are positive in the first and third quadrants.

Step 3: Analyze Quadrant 1

In the first quadrant, all trigonometric functions are positive. This suggests that it's possible for the sine, cosine, tangent, cosecant, secant, and cotangent of an angle to be positive at the same time.

Step 4: Verify Through Examples

Consider an angle like \( 45^{\circ} \), which is in the first quadrant:- \( ext{sin}(45^{\circ}) = rac{\sqrt{2}}{2} > 0 \)- \( ext{cos}(45^{\circ}) = rac{\sqrt{2}}{2} > 0 \)- \( ext{tan}(45^{\circ}) = 1 > 0 \)- \( ext{csc}(45^{\circ}) = \sqrt{2} > 0 \)- \( ext{sec}(45^{\circ}) = \sqrt{2} > 0 \)- \( ext{cot}(45^{\circ}) = 1 > 0 \)Since each function is positive, such an angle exists, confirming the possibility.

Step 5: Conclusion and Answer the Question

Based on the analysis and example provided, there is at least one angle where all six trigonometric functions are positive. This indicates that the statement is true.

Key Concepts

Sine and Cosine Functions

The sine and cosine functions are fundamental to trigonometry, representing the vertical and horizontal coordinates of a point on a unit circle, respectively.

• Sine Function (sin): The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. In the unit circle, it's the y-coordinate of the point where the terminal side of the angle intersects the circle.
• Cosine Function (cos): For cosine, the angle's value in a right triangle is the ratio of the adjacent side length to the hypotenuse. On the unit circle, it's the x-coordinate of the intersection point.

Understanding which quadrants these functions are positive or negative in helps in solving trigonometric problems. For sine, it is positive in both the first and second quadrants. Cosine is positive in the first and fourth quadrants.

In the case of an angle like 45°, located in the first quadrant, both sine and cosine are positive with a value of \( \frac{\sqrt{2}}{2} \). These values reflect that in the first quadrant, the coordinates (and thus, sine and cosine) remain positive.

Tangent and Cotangent Functions

The tangent and cotangent functions describe the ratio of the sine to cosine, and the cosine to sine, respectively, of a given angle.

• Tangent Function (tan): Tangent is calculated as the ratio of the sine value to the cosine value. It measures the slope of the angle's terminal side relative to the x-axis.
• Cotangent Function (cot): Cotangent is simply the inverse of the tangent, offering the ratio of the cosine to sine.

Both tangent and cotangent can be positive or negative depending on the angle’s quadrant.

• In the first quadrant, both functions are positive because both sine and cosine are positive, making their ratio positive as well.
• Likewise, they are positive in the third quadrant, where both sine and cosine are negative, resulting in a positive ratio.

For example, at 45° in the first quadrant, tangent equals 1 and cotangent also equals 1. These values demonstrate that in certain quadrants, like the first, all functions remain positive.

Cosecant and Secant Functions

Cosecant and secant are the reciprocal functions of sine and cosine respectively, providing an alternative way to express these classical functions.

• Cosecant Function (csc): Cosecant is defined as the reciprocal of sine, or \( \frac{1}{\sin} \). It reflects the relationship between the hypotenuse and the opposite side in a right triangle.
• Secant Function (sec): Similarly, secant is the reciprocal of cosine, calculated as \( \frac{1}{\cos} \), representing the ratio of the hypotenuse to the adjacent side.

The positivity of these functions aligns with that of sine and cosine:

• Cosecant is positive in the quadrants where sine is positive, specifically the first and second quadrants.
• Secant is positive in areas where the cosine value is positive, notably the first and fourth quadrants.

Consider an angle like 45°. In this instance, cosecant equals \( \sqrt{2} \) and secant also equals \( \sqrt{2} \), clearly demonstrating that in the first quadrant, both functions maintain positivity. This supports that all trigonometric functions are positive within the first quadrant.