# What are the first two positive x-intercepts for the given sinusoidal function?

• MHB
• mathuravasant
In summary, the first two positive x-intercepts for y= -2cos(3(x-25°)) +1 are at x = 45 and x = 65 degrees. These values can be found by setting cos(3(x-25)) equal to 1/2 and solving for x, which results in an equilateral triangle with angles of 60 degrees.
mathuravasant
Find the first two positive x-intercepts for y= -2cos(3(x-25°)) +1
(Can someone help me for this)

mathuravasant said:
Find the first two positive x-intercepts for y= -2cos(3(x-25°)) +1
(Can someone help me for this)
Well, some basic algebra first:
$$\displaystyle 0 = -2 ~ cos(3 (x - 25) ) + 1$$

$$\displaystyle -1 = -2 ~ cos(3 (x - 25) )$$

$$\displaystyle \dfrac{1}{2} = cos(3 (x - 25) )$$

Where are the first two places you find cos(y) = 1/2?

Can you finish?

-Dan

im not sure if I did it right but would the answer be 1452.4 degree +k*120 degree , 1532.4 degree +k*120 degree ??

No. Even if the numbers were right where are you getting the k's from?

Start here: Where are the first two places you find cos(y) = 1/2? If you need to use your calculator! Graph it! Do something. I have no idea how you got your answer so I can't address what you might have done wrong. Please tell us how you are doing your calculations.

The first two places you find cos(y) = 1/2 are y = 30 and y = 300 degrees. So solve 3(x - 25) = 30 and 3(x - 25) = 300.

-Dan

Cosine is "near side over hypotenuse". So if cos(y)= 1/2 we can represent it as a right triangle with hypotenuse of length 2 and one leg of length 1. Two of those right triangles, placed together gives a triangle with two sides of length 2 and the third side of length 1+ 1= 2. That's an equilateral triangle! Its three angles, and in particular the one whose cosine is 1/2, are all 60 degrees.

Since cos(3(x-25))= 1/2, 3(x- 25)= 3x- 75= 60 so 3x= 60+ 75= 135, x= 45 degrees.
Since cos(180- x)= COS(x), cos(180- 60)= cos(120)= 1/2 so we can also have 3x- 75= 120, 3x= 195, x= 65 degrees.

## 1. What is a sinusoidal function?

A sinusoidal function is a mathematical function that describes a repetitive pattern that resembles a sine or cosine curve. It is typically written as f(x) = A sin (Bx + C) + D, where A, B, C, and D are constants that determine the amplitude, period, phase shift, and vertical shift of the function, respectively.

## 2. What is the period of a sinusoidal function?

The period of a sinusoidal function is the distance between two consecutive peaks or troughs of the function. It can be calculated using the formula T = 2π/B, where B is the coefficient of x in the function. The period determines how often the function repeats itself.

## 3. How do you graph a sinusoidal function?

To graph a sinusoidal function, you can plot a few points using the given values of x and f(x), and then connect the points with a smooth curve. Alternatively, you can use the amplitude, period, phase shift, and vertical shift to determine key points on the graph and then draw the curve. It is also helpful to know that the graph of a sinusoidal function is symmetric about the line x = C.

## 4. What is the amplitude of a sinusoidal function?

The amplitude of a sinusoidal function is the distance between the center line and the peak or trough of the function. It is equal to half the distance between the maximum and minimum values of the function. The amplitude determines the vertical stretch or compression of the function.

## 5. How are sinusoidal functions used in real life?

Sinusoidal functions are used to model various natural phenomena, such as sound waves, light waves, and planetary motion. They are also commonly used in fields such as engineering, physics, and economics to analyze and predict periodic patterns and trends. For example, the stock market can be modeled using sinusoidal functions to predict fluctuations in stock prices over time.

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