The Function \[$ F \$\] Is Given By \[$ F(t) = \sin^2 T - 1 \$\].For How Many Values Of \[$ T \$\] Does \[$ F(t) = 0 \$\]?A. None B. One C. Two D. Infinitely Many
Introduction
In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. The function f(t) = sin^2(t) - 1 is a trigonometric function that involves the sine of an angle t. In this article, we will explore the roots of this function, which are the values of t that make f(t) equal to zero.
Understanding the Function
The function f(t) = sin^2(t) - 1 can be rewritten as f(t) = (sin(t))^2 - 1. This function involves the square of the sine of an angle t, which is always non-negative. Therefore, the function f(t) is always non-positive, since it is equal to the negative of a non-negative number.
Finding the Roots of the Function
To find the roots of the function f(t), we need to find the values of t that make f(t) equal to zero. This means that we need to solve the equation sin^2(t) - 1 = 0.
Solving the Equation
To solve the equation sin^2(t) - 1 = 0, we can start by adding 1 to both sides of the equation, which gives us sin^2(t) = 1. This equation can be rewritten as (sin(t))^2 = 1.
Using Trigonometric Identities
We can use the trigonometric identity sin^2(t) + cos^2(t) = 1 to rewrite the equation (sin(t))^2 = 1 as cos^2(t) = 0.
Solving for Cos(t)
Since cos^2(t) = 0, we know that cos(t) = 0. This means that the angle t must be a multiple of 90 degrees, or π/2 radians.
Finding the Values of t
Since cos(t) = 0, we know that t = π/2 + kπ, where k is an integer. This means that there are infinitely many values of t that satisfy the equation f(t) = 0.
Conclusion
In conclusion, the function f(t) = sin^2(t) - 1 has infinitely many roots, which are the values of t that make f(t) equal to zero. These roots are given by the equation t = π/2 + kπ, where k is an integer.
The Final Answer
The final answer is D. Infinitely many.
Additional Information
It's worth noting that the function f(t) = sin^2(t) - 1 is a periodic function, which means that it repeats itself over a certain interval. In this case, the function repeats itself every 2Ï€ radians. This means that the roots of the function are also periodic, and they repeat themselves every 2Ï€ radians.
Graph of the Function
The graph of the function f(t) = sin^2(t) - 1 is a periodic function that oscillates between -1 and 0. The graph has infinitely many roots, which are the values of t that make f(t) equal to zero.
Table of Roots
| t | f(t) |
|---|---|
| π/2 | 0 |
| 3Ï€/2 | 0 |
| 5Ï€/2 | 0 |
| ... | ... |
Conclusion
Q: What is the function f(t) = sin^2(t) - 1?
A: The function f(t) = sin^2(t) - 1 is a trigonometric function that involves the sine of an angle t. It is a periodic function that oscillates between -1 and 0.
Q: What are the roots of the function f(t) = sin^2(t) - 1?
A: The roots of the function f(t) = sin^2(t) - 1 are the values of t that make f(t) equal to zero. These roots are given by the equation t = π/2 + kπ, where k is an integer.
Q: How many roots does the function f(t) = sin^2(t) - 1 have?
A: The function f(t) = sin^2(t) - 1 has infinitely many roots.
Q: What is the period of the function f(t) = sin^2(t) - 1?
A: The period of the function f(t) = sin^2(t) - 1 is 2Ï€ radians.
Q: What is the graph of the function f(t) = sin^2(t) - 1 like?
A: The graph of the function f(t) = sin^2(t) - 1 is a periodic function that oscillates between -1 and 0.
Q: Can you provide a table of roots for the function f(t) = sin^2(t) - 1?
A: Yes, here is a table of roots for the function f(t) = sin^2(t) - 1:
| t | f(t) |
|---|---|
| π/2 | 0 |
| 3Ï€/2 | 0 |
| 5Ï€/2 | 0 |
| ... | ... |
Q: How do you solve the equation sin^2(t) - 1 = 0?
A: To solve the equation sin^2(t) - 1 = 0, you can start by adding 1 to both sides of the equation, which gives you sin^2(t) = 1. This equation can be rewritten as (sin(t))^2 = 1.
Q: What trigonometric identity can be used to solve the equation (sin(t))^2 = 1?
A: The trigonometric identity sin^2(t) + cos^2(t) = 1 can be used to solve the equation (sin(t))^2 = 1.
Q: How do you use the trigonometric identity sin^2(t) + cos^2(t) = 1 to solve the equation (sin(t))^2 = 1?
A: You can use the trigonometric identity sin^2(t) + cos^2(t) = 1 to rewrite the equation (sin(t))^2 = 1 as cos^2(t) = 0.
Q: What does cos^2(t) = 0 mean?
A: cos^2(t) = 0 means that cos(t) = 0.
Q: What does cos(t) = 0 mean?
A: cos(t) = 0 means that the angle t must be a multiple of 90 degrees, or π/2 radians.
Q: How do you find the values of t that satisfy the equation cos(t) = 0?
A: You can find the values of t that satisfy the equation cos(t) = 0 by using the equation t = π/2 + kπ, where k is an integer.
Q: What is the final answer to the problem of finding the roots of the function f(t) = sin^2(t) - 1?
A: The final answer is D. Infinitely many.