Find the formula for the trigonometric function

Question

Hers the link of the graph: 

 

https://s.yimg.com/hd/answers/i/0fbb4a875852440fbbfbf9a60e7ef4aa_A.png?a=answers&mr=0&x=1409721573&s=11b2b96432cd8321fe1c0d6e231884ec

 

 

Thanks

 

Answer 1

The graph of your function g looks like it has the same shape as the first "hump" of the graph of the sin function, i.e., the graph of sin that runs from θ=0 to θ=π. Let's build the formula for g by stretching the graph of the sin function so that it matches the graph of g.

The sin function reaches a maximum height of y=1. But your function g seems to reach a maximum height of y=8: g goes 8 times as high as sin. So we have to stretch the graph of the sin function vertically by a factor of 8 in order to match the graph of g. So a first stab at writing the formula for g would look like

g(θ) = 8sin(θ).

But we're not done yet, because there's another difference between sin and g. Sin goes from sea level all the way up to its maximum as θ runs from θ=0 to θ=π/2 (π/2 is about 1.5). But g goes from sea level all the way up to its maximum as θ goes from θ=0 to θ=5. So we have to stretch the graph of the sin function horizontally by a factor of 5/(π/2) = 10/π (that's about 3) in order to match the graph of g. So the finished formula for g is

g(θ) = 8sin(πθ/10)

Note that when the θ in the above formula runs from θ=0 to θ=5, the value of the argument of the sin in the above formula (i.e., the expression πθ/10) will run from πθ/10=0 to πθ/10=π/2. And when its argument runs from 0 to π/2, the sin function in the above formula will go from sea level all the way up to its maximim. So the above formula for g does exactly what we want. Thanks.