Sketch An Angle { \theta$}$ In Standard Position Such That { \theta$}$ Has The Least Positive Measure, And The Given Point Is On The Terminal Side Of { \theta$}$. Then Find The Values Of The Six Trigonometric Functions For
Introduction
In this article, we will explore the concept of sketching an angle in standard position and finding the values of the six trigonometric functions. We will start by defining what an angle in standard position is and then proceed to sketch an angle with the least positive measure. Finally, we will find the values of the six trigonometric functions for the given angle.
What is an Angle in Standard Position?
An angle in standard position is an angle that is measured counterclockwise from the positive x-axis to the terminal side of the angle. The terminal side of an angle is the side of the angle that is not on the x-axis. The angle is said to be in standard position if its vertex is at the origin (0, 0) and its initial side is on the positive x-axis.
Sketching an Angle in Standard Position
To sketch an angle in standard position, we need to follow these steps:
- Draw the positive x-axis on a coordinate plane.
- Draw the positive y-axis on the same coordinate plane.
- Draw the terminal side of the angle, which is the side of the angle that is not on the x-axis.
- Label the angle with the measure of the angle.
Sketching an Angle with the Least Positive Measure
To sketch an angle with the least positive measure, we need to draw an angle that is as small as possible. The least positive measure of an angle is 0°. To draw an angle with a measure of 0°, we need to draw a line that is parallel to the x-axis.
Finding the Values of the Six Trigonometric Functions
The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. These functions are defined as follows:
- Sine: sin(θ) = opposite side / hypotenuse
- Cosine: cos(θ) = adjacent side / hypotenuse
- Tangent: tan(θ) = opposite side / adjacent side
- Cotangent: cot(θ) = adjacent side / opposite side
- Secant: sec(θ) = hypotenuse / adjacent side
- Cosecant: csc(θ) = hypotenuse / opposite side
To find the values of the six trigonometric functions for a given angle, we need to know the values of the opposite side, adjacent side, and hypotenuse.
Finding the Values of the Opposite Side, Adjacent Side, and Hypotenuse
To find the values of the opposite side, adjacent side, and hypotenuse, we need to use the Pythagorean theorem. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the opposite side and adjacent side.
Example
Let's say we have an angle with a measure of 30°. To find the values of the six trigonometric functions for this angle, we need to know the values of the opposite side, adjacent side, and hypotenuse.
Using the Pythagorean theorem, we can find the values of the opposite side and adjacent side as follows:
- Opposite side: 1
- Adjacent side: 1/2
- Hypotenuse: √(1^2 + (1/2)^2) = √(1 + 1/4) = √(5/4) = √5/2
Now that we have the values of the opposite side, adjacent side, and hypotenuse, we can find the values of the six trigonometric functions as follows:
- Sine: sin(30°) = opposite side / hypotenuse = 1 / (√5/2) = 2 / √5
- Cosine: cos(30°) = adjacent side / hypotenuse = 1/2 / (√5/2) = 1 / √5
- Tangent: tan(30°) = opposite side / adjacent side = 1 / (1/2) = 2
- Cotangent: cot(30°) = adjacent side / opposite side = 1/2 / 1 = 1/2
- Secant: sec(30°) = hypotenuse / adjacent side = (√5/2) / (1/2) = √5
- Cosecant: csc(30°) = hypotenuse / opposite side = (√5/2) / 1 = √5/2
Conclusion
In this article, we have explored the concept of sketching an angle in standard position and finding the values of the six trigonometric functions. We have defined what an angle in standard position is and then proceeded to sketch an angle with the least positive measure. Finally, we have found the values of the six trigonometric functions for a given angle using the Pythagorean theorem.
References
- "Trigonometry" by Michael Corral
- "Precalculus" by James Stewart
- "Calculus" by Michael Spivak
Further Reading
- "Trigonometry for Dummies" by Mary Jane Sterling
- "Precalculus: Mathematics for Calculus" by James Stewart
- "Calculus: Early Transcendentals" by James Stewart
Frequently Asked Questions (FAQs) about Sketching an Angle in Standard Position and Finding Trigonometric Functions =============================================================================================
Q: What is an angle in standard position?
A: An angle in standard position is an angle that is measured counterclockwise from the positive x-axis to the terminal side of the angle. The terminal side of an angle is the side of the angle that is not on the x-axis.
Q: How do I sketch an angle in standard position?
A: To sketch an angle in standard position, you need to follow these steps:
- Draw the positive x-axis on a coordinate plane.
- Draw the positive y-axis on the same coordinate plane.
- Draw the terminal side of the angle, which is the side of the angle that is not on the x-axis.
- Label the angle with the measure of the angle.
Q: What is the least positive measure of an angle?
A: The least positive measure of an angle is 0°. To draw an angle with a measure of 0°, you need to draw a line that is parallel to the x-axis.
Q: How do I find the values of the six trigonometric functions?
A: To find the values of the six trigonometric functions, you need to know the values of the opposite side, adjacent side, and hypotenuse. You can use the Pythagorean theorem to find these values.
Q: What is the Pythagorean theorem?
A: The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the opposite side and adjacent side.
Q: How do I use the Pythagorean theorem to find the values of the opposite side, adjacent side, and hypotenuse?
A: To use the Pythagorean theorem, you need to follow these steps:
- Draw a right triangle with the angle you want to find the trigonometric functions for.
- Label the opposite side, adjacent side, and hypotenuse.
- Use the Pythagorean theorem to find the values of the opposite side, adjacent side, and hypotenuse.
Q: What are the six trigonometric functions?
A: The six trigonometric functions are:
- Sine: sin(θ) = opposite side / hypotenuse
- Cosine: cos(θ) = adjacent side / hypotenuse
- Tangent: tan(θ) = opposite side / adjacent side
- Cotangent: cot(θ) = adjacent side / opposite side
- Secant: sec(θ) = hypotenuse / adjacent side
- Cosecant: csc(θ) = hypotenuse / opposite side
Q: How do I find the values of the six trigonometric functions for a given angle?
A: To find the values of the six trigonometric functions for a given angle, you need to know the values of the opposite side, adjacent side, and hypotenuse. You can use the Pythagorean theorem to find these values.
Q: What is the difference between the sine and cosine functions?
A: The sine function is the ratio of the opposite side to the hypotenuse, while the cosine function is the ratio of the adjacent side to the hypotenuse.
Q: What is the difference between the tangent and cotangent functions?
A: The tangent function is the ratio of the opposite side to the adjacent side, while the cotangent function is the ratio of the adjacent side to the opposite side.
Q: What is the difference between the secant and cosecant functions?
A: The secant function is the ratio of the hypotenuse to the adjacent side, while the cosecant function is the ratio of the hypotenuse to the opposite side.
Conclusion
In this article, we have answered some of the most frequently asked questions about sketching an angle in standard position and finding trigonometric functions. We hope that this article has been helpful in clarifying any confusion you may have had about these topics. If you have any further questions, please don't hesitate to ask.
References
- "Trigonometry" by Michael Corral
- "Precalculus" by James Stewart
- "Calculus" by Michael Spivak
Further Reading
- "Trigonometry for Dummies" by Mary Jane Sterling
- "Precalculus: Mathematics for Calculus" by James Stewart
- "Calculus: Early Transcendentals" by James Stewart