sinusoidal

functions and trigonometry

Table of contents

pg 3: introduction

pg 4-11: chapter 1 and sine law

pg 12- 19: chapter 2 and cosine law

pg 20- 30: chapter 3 and sinusoidal functions

pg 31- 45: chapter 4 and creating a graph

This is Daisy the dog. Daisy escaped her house in Doglandia and is on a journey in the city to find her way back to her home using Sinusoidal functions and Trigonometry!

I'm Daisy the dog, help me find my way back home!

chapter 1: daisy the hungry dog

In Doglandia, all the food is located at the end of the rainbow, this place is called Dog Food Heaven and is where all the dogs go to have the best meal ever!

i see the rainbow! I am so hungry

C

A

B

70

65

20m

18m

?

To find out how far i am from Dog Food Heaven we must use sine law to find side a

How to solve : Sine Law

step 1: determine what you have

angle A= 65 degrees angle B= 70 degrees angle C= ?

side A= ? side b= 20m side C= 18m

note: all angles in a triangle MUST add up to be 180 degrees, therefore to find a missing angle we must subtract the two KNOWN angles from 180 to determine our missing angle.

180 - 70 - 65 = 45 degrees

therefore we have determined that Angle C is 45 degrees

step 2: determine what formula you must use

note: sine law uses 2 formulas, one is for finding angles, and the other is for finding side lengths

sin A = sin B = sin C

a b c

note: this is the formula to find angles

a = b = c

sin A. sin B sin C

note: this is the formula we will be using since, this is the formula used to find side lenghts

step 3: input the numbers in the equation

we will be using a, since it is what we are trying to find, and b since we are given all the information

a = 20

sin 65 sin 70

(sin 65)

(sin 65)

note: each side needs to be multiplied by Sin 65 so that we get " a " by itself, and whatever we do to one side we must do to the other

a = 20

sin 70

(sin 65)

now we are left with this since on the left side sin 65 was canceled out because we multiplied it by itself.

note: an important thing to note is to keep all the decimals to get an accurate answer

a= (21.28355544951824) (0.90630778703665)

you must multiply these two numbers to get your final answer

therefore, using sine law, we have determined that side a is equal to 19.3m

a = 19.3m

Daisy the dog finally made it to Dog Food Heaven

using sine law i was able to find out how far i was from dog food heaven and now i can enjoy this delicious food!

chapter 2: The bone

Oh my gosh a dog bone! My family would love that.

on her journey home, Daisy spotted a dog bone up in a tree and wanted to bring it home for her family and friends to enjoy

i need to jump up to get the bone from the tree, but i can only jump at a 65 degree angle. i need to figure out how far up i need to jump using cosine law to reach the bone.

55

8.5m

11m

A

B

C

39

How to solve: Cosine Law

we must use cosine law to determine how far up Daisy must jump at a 55 degree angle to reach the bone

notes to keep in mind when using cosine law:

• we can use this when we are given all 3 sides or 2 sides and the included angle

note: cosine is the best option to use since we are given 2 sides and the corresponding angle

the formula we will be using to solve this problem is:

b^2 = a^2 + c^2 - 2ac x cosB

Step 1: determine what you have

angle B = 65 degrees

side A = ? side B = 8.5m side C = 11m

step 2: plug the numbers into the formula

b^2 = 8.5^2 + 11^2 - 2(8.5)(11) x cos39

in order to simplify the equation, calculate the square root of A and C

b^2 = 72.25 + 121 - 2(8.5)(11) x cos39

step 3: add the first set of numbers and simplify the second part of the equation

b^2 = 193.25 - 145.3262947924536

next subtract the two numbers from each other

b^2 = 47.9237052075464

take the square root of b^2 and the square root of 47.923.... to get your final answer!

b= 6.9m

Therefore, Daisy the dog must jump 6.9m at a 55 degree angle to get the bone!

Thanks to the Cosine Law, Daisy the dog was able to find out how high she must jump to reach the bone!

i am so happy i got my bone!

chapter 3: doglandia beach

On her journey home, Daisy came across Doglandia's most beautiful beaches called

Dogtopia

Dogtopia is so beautiful! i have never seen it before, look at all these beautiful waves aswell!

Before daisy leaves the beach, she wants surf on the big waves. Daisy finds a lifeguard that tells her the information on the waves since its her first time

excuse me lifeguard can you help me?

yes, for sure! i will tell you everything

The lifeguard tells Daisy that when she gets on the wave she will be at a height of 1m at 0 seconds, and a maximum height of 6m at 10 seconds. Daisy wants to know how high she will be at 5 seconds on the wave

to help me out with my problem, i must draw out a model to help me. Hmmm where should i write it? i know! i will write it in the sand

How to solve: Sinusoidal function

the formula that we will be using throughout this problem is

y = a sin (k (x-d) + c

note : we will be using various formulas throughout this problem to solve for our values. once that is solved we will use that information to find out how high Daisy will be at 5 seconds.

step 1: calculate the A value which represents the Amplitude

To determine the amplitude we must take our maximum value which is 6 and subtract it from our y value which is our equation of the axis (EOA)

first we need to find our EOA:

maximum + minimum

2

= 6 + 1 EOA= 3.5

2

Amplitude:

maximum - EOA

6 - 3.5 = 2.5

therefore, our amplitude is 2.5

step 2: Now we must calculate our K value, which represents our horizontal compression

k = 360

period

We know that our period is equal to 10 seconds

k = 360 k= 36

10

our graph does not shift since we start at 0, therefore our D value which represents horizontal translation is 0

step 3: now that we have solved for all of our values, we can plug it into the equation for 5 seconds!

recall our formula:

y = a sin (k (x-d) + C

input all of your values into the formula

y= 2.5 sin (36(5-0) + 0

y= 2.5 sin (180)

now we must calculate the cos of 180

y = 2.5 + 0

y= 2.5 m

therefore at 5 seconds surfing daisy will be at a height of 2.5 meters