Problem 6 - Precalculus

Question

The terminal side of an angle \(\theta\) in standard position passes through values of the six trigonometric functions for angle \(\theta\) $$(-1,3)$$

Step-by-Step Solution

Answer

The values of the trigonometric functions are: \( \sin(\theta) = \frac{3\sqrt{10}}{10} \), \( \cos(\theta) = \frac{-\sqrt{10}}{10} \), \( \tan(\theta) = -3 \), \( \csc(\theta) = \frac{\sqrt{10}}{3} \), \( \sec(\theta) = -\sqrt{10} \), \( \cot(\theta) = \frac{-1}{3} \).

Step 1: Determine the Radius

Calculate the radius (hypotenuse) using the Pythagorean theorem for the point \(-1, 3\): \[ r = \sqrt{(-1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \]

Step 2: Calculate the Sine Function

The sine of angle \( \theta \) is the ratio of the y-coordinate to the hypotenuse: \[ \sin(\theta) = \frac{3}{\sqrt{10}} \] Rationalize the denominator: \[ \sin(\theta) = \frac{3\sqrt{10}}{10} \]

Step 3: Calculate the Cosine Function

The cosine of angle \( \theta \) is the ratio of the x-coordinate to the hypotenuse: \[ \cos(\theta) = \frac{-1}{\sqrt{10}} \] Rationalize the denominator: \[ \cos(\theta) = \frac{-\sqrt{10}}{10} \]

Step 4: Calculate the Tangent Function

The tangent of angle \( \theta \) is the ratio of the sine to the cosine, or y-coordinate to x-coordinate:\[ \tan(\theta) = \frac{3}{-1} = -3 \]

Step 5: Calculate the Cosecant Function

The cosecant is the reciprocal of the sine function:\[ \csc(\theta) = \frac{\sqrt{10}}{3} \]

Step 6: Calculate the Secant Function

The secant is the reciprocal of the cosine function:\[ \sec(\theta) = -\sqrt{10} \]

Step 7: Calculate the Cotangent Function

The cotangent is the reciprocal of the tangent function:\[ \cot(\theta) = \frac{-1}{3} \]

Key Concepts

Sine Function

The sine function is one of the most important functions in trigonometry. It represents the y-coordinate of a point on the unit circle, which helps us understand the angle's position. The sine of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle.
Given a specific problem, if we know the coordinates of a point, the sine function can be calculated using the ratio \( \frac{y}{r} \), where \( y \) is the y-coordinate, and \( r \) is the hypotenuse or the radius of the circle.

• The sine function is essential because it tells us how far "up" or "down" the angle is on the unit circle.
• It can take values from -1 to 1, depending on the angle \( \theta \).
• In our problem, the sine of angle \( \theta \) is \( \frac{3}{\sqrt{10}} \), which simplifies to \( \frac{3\sqrt{10}}{10} \).

This expression indicates that the angle \( \theta \) is above the horizontal axis, given that the y-coordinate is positive.

Cosine Function

The cosine function is another fundamental trigonometric function. It reflects a point's x-coordinate on the unit circle, crucial in determining the angle's placement horizontally. For any angle \( \theta \), the cosine is the ratio of the adjacent side's length to the hypotenuse in a right triangle.
We calculate it using the fraction \( \frac{x}{r} \), wherein this context, \( x \) is the x-coordinate, and \( r \) is the circle's radius. In trigonometric terms, cosine helps pinpoint how far "left" or "right" the angle is positioned from the vertical axis.

• The cosine function also varies between -1 and 1.
• For our problem, the cosine of angle \( \theta \) is \( \frac{-1}{\sqrt{10}} \), which converts to \( \frac{-\sqrt{10}}{10} \).
• This shows that the angle \( \theta \) is positioned to the left of the origin, as indicated by a negative x-coordinate.

Cosine values are particularly useful in understanding the horizontal aspect of an angle's position on a circle or coordinate plane.

Tangent Function

The tangent function connects both sine and cosine by offering a ratio between them. It represents the slope of the line formed by the terminal side of an angle \( \theta \) and the horizontal axis.
In trigonometric terms, tangent is described as \( \frac{\sin(\theta)}{\cos(\theta)} \), or simply, the ratio of the y-coordinate to the x-coordinate. It's significantly helpful in questions involving angles of elevation and depression.

• The tangent function can have any real number as its value, but it becomes undefined when the cosine is zero.
• For the exercise problem, \( \tan(\theta) \) is \( \frac{3}{-1} = -3 \).
• This negative value suggests a downward slope, which means the angle is moving downwards as it comes from the positive y-axis direction.

Understanding tangent helps us analyze the relative steepness or slope of the angle in view, which is pivotal for practical aspects of geometry and physics.