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Magazine Chapter 8: Problem 3
Evaluate the trigonometric functions at the angle (in standard position) whose terminal side contains the given point. $$(-3,7)$$
Short Answer
Expert verified
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(伪) = 7/鈭�58, cos(伪) = -3/鈭�58, tan(伪) = -7/3.
Step by step solution
01
Find the reference angle
We have the coordinates of a point #(-3, 7)# on the terminal side of the angle. The reference angle \(胃\) is the angle formed by the positive x-axis and the terminal side of the angle. To find \(胃\), we'll use the arctangent function:
$$胃 = arctan\left(\frac{y}{x}\right) = arctan\left(\frac{7}{-3}\right)$$
Here, x = -3 and
y = 7.
02
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 蟺 in radians) to the reference angle:
$$伪 = 胃 + 蟺 = arctan\left(\frac{7}{-3}\right) + 蟺$$
03
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 \(r\):
$$r = \sqrt{(-3)^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58}$$
04
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:
$$sin(伪) = \frac{y}{r} = \frac{7}{\sqrt{58}}$$
2. Cosine function:
$$cos(伪) = \frac{x}{r} = \frac{-3}{\sqrt{58}}$$
3. Tangent function:
$$tan(伪) = \frac{y}{x} = \frac{7}{-3} = -\frac{7}{3}$$
So, the values of the trigonometric functions at the angle are:
$$sin(伪) = \frac{7}{\sqrt{58}}, cos(伪) = \frac{-3}{\sqrt{58}}, tan(伪) = -\frac{7}{3}$$
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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 \( \theta = \arctan\left(\frac{7}{3}\right) \).
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:
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.
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:
- \( r = \sqrt{(-3)^2 + 7^2} \)
- \( r = \sqrt{9 + 49} = \sqrt{58} \)
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, \( \tan(\alpha) = \frac{y}{x} \).
- \( \sin(\alpha) = \frac{7}{\sqrt{58}} \)
- \( \cos(\alpha) = \frac{-3}{\sqrt{58}} \)
- \( \tan(\alpha) = -\frac{7}{3} \)
