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# Trigonometry: FAQ

## What is the unit circle and why is it important in trigonometry?

The unit circle is a circle with a radius of 1 that is centered at the origin on a coordinate plane. It's important in trigonometry because it allows us to define the sine and cosine functions in terms of the x- and y-coordinates of a point moving around the circle.

## What are radians and why do we use them in trigonometry?

Radians are a unit of measurement for angles.
One radian is the angle measure that we turn to travel one radius length around the circumference of a circle.
We often use radians in trigonometry because they make working with trigonometric functions easier.

## What is the Pythagorean identity and why is it important?

The Pythagorean identity is sine, squared, x, plus, cosine, squared, x, equals, 1. It comes from the Pythagorean theorem and is important in trigonometry because it can help us solve for the value of one trigonometric function if we know the other.

## What do amplitude, midline, and period mean when we're talking about sinusoidal graphs?

The amplitude of a sinusoidal graph is the distance from the midline to the highest or lowest point on the graph. The midline is the horizontal line that the graph oscillates around, and the period is the horizontal distance it takes for the graph to complete one full cycle.

## Why do we want to know how to transform sinusoidal graphs?

By understanding how to transform sinusoidal graphs, we can graph a wider variety of sinusoidal functions. For example, we can change the amplitude, midline, or period to match a given equation.

## Where are trigonometric functions used in the real world?

Trigonometric functions are used in many real-world applications. For example, engineers use them to design bridges, and physicists use them to model periodic phenomena such as waves or vibrations.

## Want to join the conversation?

• Did the modeling w/ sinusoidal functions (phase shift) unit; how to know when to do sin or cos since either can happen w/ a phase shift? Instructions don't specify which/my answer satisfies requirements, but it isn't counted as correct. • As we know, when looking at a unit circle, cosine's value will reach 1 when x=0 radians, whereas sine's value will reach 1 when x=pi/2 radians. In a scenrio where the graph reaches its maximum point when x=0, you know that it will be a graph of cosine. However, if you have a scenario where you reach the midline (midpoint) at a x-value of 0, you know you have a graph of sine. Essentially, you want to look at where you reach your maximum or midline when analyzing what type of a graph to use. Now, if you are given a problem with a phase shift, you want to look at the maximum and minium and shift it with relevance to what format (sine or cousin) your graph is in. I would suggest looking at one of the videos that Sal posted. He really thoroughly explains how to know what format to use.
• I am struggling to understand one of the problems I encountered where I was asked WHEN the function would hit it's maximum.

There was a function given,

h(t) = 5 - 2 * sin(2pi (t + 1) / 7)

I understand that the function is negative so the it's a sine reflection.

I think I also understand that the function ignoring the +1 would reach it's maximum at t=5.25 (i.e. 3/4 * 7)

what I don't understand is why when considering the +1 the max takes place at t=6.25 instead of t=4.25?

Shouldn't a +1 SHIFT the whole function to the LEFT - causing the MAX to be encountered sooner by 1?

Thank you. • I really do not understand the substitution in phase shifting a function. I've tried it in every way I can imagine solving it... for example: −1.5cos(2π/248(22+11))+5.9 , solved in several different ways, always evaluated to approx. 4.4002.... I got those questions wrong every single time. what am I doing wrong here? what am I missing?? • could you show me your work? what are the steps you take to get this answer? Also, what is the problem? This is how I would solve it:

1)(33π/248) ≈ 0.836069...

2) cos(0.836069) ≈ 0.6703848... MAKE SURE TO USE RADIAN MODE FOR THIS STEP. The khan academy calculator(and most others) have a deg and rad mode for trig functions.

3) -1.5*0.670384... ≈ -1.0055..

4) -1.0055...+5.9 = 4.894422...

Hope this helped! Next time show your work as well :)    