Problem 3 - Contemporary Precalculus

Question

Evaluate the trigonometric functions at the angle (in standard position) whose terminal side contains the given point. $$(-3,7)$$

Step-by-Step Solution

Answer

Question: Given a point (-3, 7) on the terminal side of an angle in standard position, find the values of sine, cosine, and tangent of the angle. Answer: sin(α) = 7/√58, cos(α) = -3/√58, tan(α) = -7/3.

Step 1: Find the reference angle

We have the coordinates of a point #(-3, 7)# on the terminal side of the angle. The reference angle \(θ\) is the angle formed by the positive x-axis and the terminal side of the angle. To find \(θ\), we'll use the arctangent function: $$θ = arctan\left(\frac{y}{x}\right) = arctan\left(\frac{7}{-3}\right)$$ Here, x = -3 and y = 7.

Step 2: Calculate the angle in standard position

Since the point is located in the second quadrant, where the x-coordinate is negative and the y-coordinate is positive, we can find the angle in standard position by adding 180 degrees (or π in radians) to the reference angle: $$α = θ + π = arctan\left(\frac{7}{-3}\right) + π$$

Step 3: Find the hypotenuse

Now, let's find the hypotenuse \(r\) of the right triangle formed by the terminal side of the angle, the reference angle, and the positive x-axis, using the Pythagorean theorem: $$r^2 = x^2 + y^2 = (-3)^2 + 7^2$$ Solving for \(r\): $$r = \sqrt{(-3)^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58}$$

Step 4: Evaluate the trigonometric functions

Now that we have the angle in standard position and the hypotenuse, we can find the sine, cosine, and tangent of the angle. 1. Sine function: $$sin(α) = \frac{y}{r} = \frac{7}{\sqrt{58}}$$ 2. Cosine function: $$cos(α) = \frac{x}{r} = \frac{-3}{\sqrt{58}}$$ 3. Tangent function: $$tan(α) = \frac{y}{x} = \frac{7}{-3} = -\frac{7}{3}$$ So, the values of the trigonometric functions at the angle are: $$sin(α) = \frac{7}{\sqrt{58}}, cos(α) = \frac{-3}{\sqrt{58}}, tan(α) = -\frac{7}{3}$$

Key Concepts

Reference Angle

A reference angle is the smallest positive angle formed between the terminal side of an angle and the x-axis. It is always measured in the positive direction, either upwards from the positive x-axis or downwards from the x-axis, depending on the quadrant where the angle lies.
An important feature of reference angles is that they are always between 0 and 90 degrees, or 0 and \( \frac{\pi}{2} \) radians.
When you have a point like (-3, 7), the reference angle can be found by using the arctangent function:

• The formula is \( \theta = \arctan\left(\frac{y}{|x|}\right) \).
• This accounts for the point's location relative to the x-axis.
• In our example, the reference angle \( \theta = \arctan\left(\frac{7}{3}\right) \).

The reference angle simplifies the process of finding the sine, cosine, and tangent functions since these angles share common properties.

Standard Position

When angles are in standard position, their vertex is at the origin of the Cartesian coordinate system, and their initial side lies along the positive x-axis. From there, the terminal side will rotate either clockwise or counterclockwise.
For instance, if an angle's terminal side passes through the point (-3, 7), it indicates that the angle is located in the second quadrant since the x-coordinate is negative while the y-coordinate is positive.
Understanding the basic rotation helps us:

• Identify the angle's position with respect to the reference angle.
• Recognize that, in the second quadrant, the value of an angle can be expressed by adding 180 degrees (or \( \pi \) radians) to the reference angle.
• Calculate \( \alpha = \theta + \pi \) for angles in the second quadrant.

Dotting this concept helps you to compute trigonometric function values correctly in any quadrant.

Pythagorean Theorem

The Pythagorean theorem is vital when dealing with trigonometric functions and angles. It's used to find the hypotenuse or a missing side of a right triangle. The theorem states that the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides (\(a\) and \(b\)): \[c^2 = a^2 + b^2\]For the point (-3, 7), the hypotenuse \(r\) can be found using:

• \( r = \sqrt{(-3)^2 + 7^2} \)
• \( r = \sqrt{9 + 49} = \sqrt{58} \)

The value of \( r \) is crucial because it is used as the denominator in the sine and cosine calculations. Recognizing this, you can find the size of any side or utilize it in formulating trigonometric relationships.

Sine, Cosine, and Tangent

Trigonometric functions such as sine, cosine, and tangent allow us to explore relationships within a right triangle and beyond. They are defined as follows:

• Sine (\( \sin \)): This function is the ratio of the length of the opposite side to the hypotenuse. For an angle \( \alpha \) in standard position, \( \sin(\alpha) = \frac{y}{r} \).
• Cosine (\( \cos \)): This function is the ratio of the length of the adjacent side to the hypotenuse. For an angle \( \alpha \), \( \cos(\alpha) = \frac{x}{r} \).
• Tangent (\( \tan \)): This function is the ratio of the length of the opposite side to the adjacent side. So, \( \tan(\alpha) = \frac{y}{x} \).

For the point (-3, 7):

• \( \sin(\alpha) = \frac{7}{\sqrt{58}} \)
• \( \cos(\alpha) = \frac{-3}{\sqrt{58}} \)
• \( \tan(\alpha) = -\frac{7}{3} \)

Grasping these definitions postulates understanding what these values mean in context. It enhances your ability to navigate and calculate angles and their associated trigonometric values efficiently.