Given The Ordered Pair $P(-4, 3$\], Determine The Requested Sec, Csc, Or Cot Of Angle $\theta$.$\csc \theta = $A. $\frac{5}{4}$ B. $\frac{5}{3}$ C. $-\frac{5}{3}$ D. $-\frac{5}{4}$
Trigonometric Ratios: Finding Sec, Csc, or Cot of Angle θ
In trigonometry, we often encounter problems that require us to find the values of various trigonometric ratios, such as secant, cosecant, or cotangent, given the coordinates of a point on the unit circle. In this article, we will explore how to determine the requested sec, csc, or cot of angle θ given the ordered pair P(-4, 3).
Before we dive into the problem, let's briefly review the definitions of secant, cosecant, and cotangent.
- Secant (sec): The secant of an angle θ is defined as the ratio of the length of the hypotenuse to the length of the adjacent side. It is the reciprocal of the cosine function.
- Cosecant (csc): The cosecant of an angle θ is defined as the ratio of the length of the hypotenuse to the length of the opposite side. It is the reciprocal of the sine function.
- Cotangent (cot): The cotangent of an angle θ is defined as the ratio of the length of the adjacent side to the length of the opposite side. It is the reciprocal of the tangent function.
Given the ordered pair P(-4, 3), we can use the coordinates to determine the values of the trigonometric ratios. Let's start by finding the value of the cosecant (csc) of angle θ.
Step 1: Determine the Hypotenuse
To find the value of the cosecant, we need to determine the length of the hypotenuse. We can use the Pythagorean theorem to find the length of the hypotenuse.
The Pythagorean theorem states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). In this case, we have:
c² = a² + b² c² = (-4)² + 3² c² = 16 + 9 c² = 25 c = √25 c = 5
Step 2: Determine the Opposite Side
Next, we need to determine the length of the opposite side. In this case, the opposite side is the y-coordinate of the point P, which is 3.
Step 3: Calculate the Cosecant
Now that we have the lengths of the hypotenuse and the opposite side, we can calculate the value of the cosecant.
csc θ = c / b csc θ = 5 / 3
In this article, we explored how to determine the requested sec, csc, or cot of angle θ given the ordered pair P(-4, 3). We used the Pythagorean theorem to find the length of the hypotenuse and then calculated the value of the cosecant. The final answer is:
Here are a few additional examples to help you practice finding the requested sec, csc, or cot of angle θ.
Example 1
Given the ordered pair P(3, -4), determine the value of the secant (sec) of angle θ.
Step 1: Determine the Hypotenuse
Using the Pythagorean theorem, we can find the length of the hypotenuse:
c² = a² + b² c² = 3² + (-4)² c² = 9 + 16 c² = 25 c = √25 c = 5
Step 2: Determine the Adjacent Side
Next, we need to determine the length of the adjacent side. In this case, the adjacent side is the x-coordinate of the point P, which is 3.
Step 3: Calculate the Secant
Now that we have the lengths of the hypotenuse and the adjacent side, we can calculate the value of the secant.
sec θ = c / a sec θ = 5 / 3
Example 2
Given the ordered pair P(-3, 4), determine the value of the cotangent (cot) of angle θ.
Step 1: Determine the Adjacent Side
Using the coordinates, we can determine the length of the adjacent side:
adjacent side = -3
Step 2: Determine the Opposite Side
Next, we need to determine the length of the opposite side. In this case, the opposite side is the y-coordinate of the point P, which is 4.
Step 3: Calculate the Cotangent
Now that we have the lengths of the adjacent side and the opposite side, we can calculate the value of the cotangent.
cot θ = adjacent side / opposite side cot θ = -3 / 4 cot θ = -\frac{3}{4}
In conclusion, we have explored how to determine the requested sec, csc, or cot of angle θ given the ordered pair P(-4, 3). We used the Pythagorean theorem to find the length of the hypotenuse and then calculated the value of the cosecant. The final answer is:
In our previous article, we explored how to determine the requested sec, csc, or cot of angle θ given the ordered pair P(-4, 3). In this article, we will answer some frequently asked questions related to trigonometric ratios.
Q: What is the difference between secant and cosecant?
A: The secant and cosecant are both reciprocal trigonometric ratios. The secant is the reciprocal of the cosine function, while the cosecant is the reciprocal of the sine function.
Q: How do I determine the value of the secant of an angle?
A: To determine the value of the secant of an angle, you need to know the length of the hypotenuse and the length of the adjacent side. You can use the Pythagorean theorem to find the length of the hypotenuse, and then calculate the secant using the formula:
sec θ = c / a
Q: How do I determine the value of the cosecant of an angle?
A: To determine the value of the cosecant of an angle, you need to know the length of the hypotenuse and the length of the opposite side. You can use the Pythagorean theorem to find the length of the hypotenuse, and then calculate the cosecant using the formula:
csc θ = c / b
Q: What is the cotangent of an angle?
A: The cotangent of an angle is the ratio of the length of the adjacent side to the length of the opposite side. It is the reciprocal of the tangent function.
Q: How do I determine the value of the cotangent of an angle?
A: To determine the value of the cotangent of an angle, you need to know the length of the adjacent side and the length of the opposite side. You can calculate the cotangent using the formula:
cot θ = adjacent side / opposite side
Q: What are some common trigonometric identities?
A: There are several common trigonometric identities that you should know:
- sin² θ + cos² θ = 1
- tan² θ + 1 = sec² θ
- cot² θ + 1 = csc² θ
Q: How do I use trigonometric ratios to solve problems?
A: To use trigonometric ratios to solve problems, you need to identify the given information and the unknown quantity. You can then use the appropriate trigonometric ratio to set up an equation and solve for the unknown quantity.
Q: What are some real-world applications of trigonometric ratios?
A: Trigonometric ratios have many real-world applications, including:
- Navigation: Trigonometric ratios are used in navigation to determine distances and directions.
- Physics: Trigonometric ratios are used in physics to describe the motion of objects.
- Engineering: Trigonometric ratios are used in engineering to design and build structures.
In this article, we have answered some frequently asked questions related to trigonometric ratios. We have also discussed some common trigonometric identities and real-world applications of trigonometric ratios. By understanding these concepts, you can use trigonometric ratios to solve problems and make informed decisions in a variety of fields.
If you want to learn more about trigonometric ratios, here are some additional resources:
- Khan Academy: Trigonometry
- Mathway: Trigonometric Ratios
- Wolfram Alpha: Trigonometric Ratios
In conclusion, trigonometric ratios are an essential part of mathematics and have many real-world applications. By understanding the concepts and formulas discussed in this article, you can use trigonometric ratios to solve problems and make informed decisions in a variety of fields.