**Trigonometry Expert Project**

**Trigonometric Functions**

Trigonometric functions can be used to solve for the missing sides lengths or angle measurements in any right triangle.

The most common trigonometric functions are sine, cosine, and tangent, which all relate the sides lengths of a triangle to its angles.

Tips, Tricks, & Reminders

Rotating the triangle facilitates easier visual understanding

Hypotenuses only exist in right triangles

Trigonometric ratios can only be used with right triangles, because there needs to be a hypotenuse

While given both the sides of a right triangle, and asked to find the angle, you need to use sin inverse, also known as arc sine. (Sine inverse is sine to the negative one exponent)

Sometimes, you will have to apply your knowledge of sine, cosine, and tangent to solve a multi-step trigonometry problem. A multi-step trig problem might ask you to solve for a cosine first, and then use sine to find the ultimate answer. It is basically using a combination of the three trig ratios to arrive at one final answer.

All triangles have to add up to 180

**Anvita Prabhu ~ FST: A3 ~ Miss Poluan**

**Radian**

Radian is unit of angle measurement. Just like feet, kilometers, and grams measure certain things, Radian is a way to measure angles. Radian and degree measure the same thing: angles.

One radian is the angle measurement at which the length of a radius in the a circle fits inside the circumference.

**Graphs &Transformations**

**Solving Trigonometric Equations**

There are certain types of trigonometric equations that when set equal to each other are always true, sometimes true, or never true

sin (x) = cos (x + π/2)

sin (x) = cos (x - π/2)

sin (x) + 1 = cos (x-1)

The equation can be classified as:

Always true if the graphs of the two equations overlap at all points. (When solving, there are infinitely many solutions)

Sometimes true if the graphs of the two equations intersect at one or more points. (When solving, there are only a limited solutions)

Never true if the graphs of the two equations never intersect or overlap at any points (When solving, there are no solutions)

Types of Triangles

In finding side lengths, and angles measurements, there are three categories that triangle separate into:

1. Right triangles

2. Non-right triangles

3. Special-case triangles

The sine of an angle is the ratio of the opposite side of a certain angle divided by its hypotenuse.

Sine Function

The cosine of an angle is the ratio of the adjacent side of a certain angle divided by its hypotenuse.

Cosine Function

The tangent of an angle is the ratio of the opposite side of a certain angle divided by its adjacent side.

Tangent

Right Triangles

As mentioned previously, by using sine, cosine, and tangent (SOH-CAH-TOA), it is possible to find missing sides lengths and angles of ANY right triangle.

Special Case Triangles

Within the category of right triangles, there are two types of special case triangles:

The 30-60-90 degree triangle

The 45-45-90 degree triangle

Special case triangles have specific trigonometric ratios that remain constant no matter how little or big the side lengths or angles are.

Non-Right Triangles

Finding the side lengths or angles of non-right triangles requires using either the Law of Sine or Law of Cosine

Due to the variables in the equations, whether you use the Law of Sine or Law of Cosine depends on how much information you are already given

Use Law of Sine when a given triangle has the measurements of:

Angle-Side-Angle (ASA)

OR

Angle-Angle-Side (AAS)

Use Law of Cosine when a given triangle has the measurements of:

Side-Side-Side (SSS)

OR

Side-Angle-Side (SAS)

Remember that whether one writes 30° or π/6, it means the same thing: They both mean the same exact angle measure- only written in different terms.

Keep all radians in terms of π (π is the same thing as 180°, therefore 2π is the radian of one full circle, or 360°)

Radian allows us to use easy, and exact figures (through the usage of pi), rather than "messy" decimals.

Using radians are only really effective while measuring angle measures on a (unit) circle.

Tips, Tricks, & Reminders

General Equations Graphs

By manipulating the variables in the general equation for sine and cosine waves, you are able to transform the graph

SINE GRAPH

f(x)= a sin (b (x-h)) + k

COSINE GRAPH

f(x)= a cos (b (x-h)) + k

Changing the value of the "a" & "k" change the y values of a sine/cosine wave

The change in "k" moves the sine/cosine wave up or down. The k also represents the mid line of the graph.

The "a" represents the amplitude, which is half the length from the maximum of the graph to the minimum. The change in "a" stretches a graph vertically. A negative value can flip it across the x axis.

Changing the value of the "b" & "h" change the x values of a sine/cosine wave

The change in "h" moves the sine/cosine wave right or left

The "b" represents the period, which is the length of one cycle.

The parent graph has a "b" value of 2π because 2π represents one full rotation in a circle, or 360°, so when we transform "b", we have to make sure it is in terms of 2π

There is a special equation for this:

p = 2π/b (where p is the period, 2π represents on rotation, and b represents the stretch, or modification of the rotation

Parent Graphs

Understanding and knowing the parents graphs facilitates easier understanding of how the changing of certain variables has changed the "look" of the graph, especially when writing equations for given graphs, or certain restrictions/limitations

Representations of Variables from Graphs to Unit Circles

Amplitude ("a")

Meaning on graph= half the length from the maximum point to the minimum (how much the graph has stretched vertically) (A negative "a" will flip the wave across the x axis)

Meaning on unit circle= radius

Period ("b")

Meaning on graph= how much the graph has been stretched horizontally

Meaning on unit circle= how long one cycle is and the speed of how fast or slow a rotation is going around the circle

"H"

Meaning on graph= How much the parent graph has moved left or right

Meaning on unit circle= Starting point on the unit circle

"K"

Meaning on graph= How much the graph has moved up or down

Meaning on unit circle=how much the circle's center has shifter

for sine: it goes up or down (because sine represents height)

for cosine: it does left or right (because cosine represents base)

Relationship Between Sine & Cosine

Sine and cosine graphs are the same thing, except for the fact that cosine graph are moved 90° or π/2 to the left.

Therefore, when we are writing equations, we are able to manipulate the variables in more than one way to describe one graph.

Graphs of Sine & Cosine Waves

Just as we visually represent sine and cosine on unit circles, we can do that on a four-quadrant grid as well. Taking the angle measures from the unit circle, we can "unwind" these to act as "x" values on a grid. This is known as the input. To find the "y" value, which represents the height of the function, we would solve for the output: sin(ø) = y or cosine(ø) = y. One "x" value (angle measure), and its "y" value make up a coordinate point on which the graph lies on. Sine and Cosine waves are all the same, except that the cosine wave starts shifted 90° or π/2 to the left.

* ø represents the angle measure (degree or radian (more commonly used))

The unit circle is a circle with the radius of one, the center of which is the origin (0,0) on a grid. We use the unit circle to visualize what the sine and cosine functions look like. Furthermore, as the picture will show you, the unit circle uses exact and more accurate numbers rather than decimals.

**The Unit Circle**

At a certain point, or angle of rotation, in the unit circle, sine, cosine, and tangent mean and represent different things. The starting point is at (0,0), and everything else is measured out as the distance from the starting point. The amount of rotation from the starting point, is the radian or degree value (since we know that they mean the same thing)

At angle of rotation ø:

Sine(ø) Height value

Cosine(ø) Base value

Tangent (ø) The Slope value

The height, base, and slope value are of the triangle that is formed at a point (angle of rotation) in the unit circle.

Sine, Cosine, & Tangent in a Unit Circle

So, to recap:

The equation y= sin(ø) can also be written as f(x)= sin (x)

In the unit circle:

The "x" represents the angle of rotation from the starting point 0

The "f(x)" represents the height of the triangle created at that certain angle of rotation

On the graph

The "x" represents the x values

The "f(x)" represents the y values

The equation y= cos(ø) can also be written as f(x)= cos (x)

In the unit circle:

The "x" represents the angle of rotation from the starting point 0

The "f(x)" represents the base of the triangle created at that certain angle of rotation

On the graph

The "x" represents the x values

The "f(x)" represents the y values

As you can see here, both sine and cosine representations of variables mean the same thing on the grid, they only differ in what they represent on the unit circle.

The Pythagorean Identity

The Pythagorean Identity is an equation that describes the relationship between the side lengths of a right triangle formed in a unit circle with the radius as the hypotenuse

The equation:

sin²(ø) + cosine²(ø) = 1

The Pythagorean identity is simply another way for solving for missing trigonometric values

A point on the unit circle can be classified as (cosineø, sineø), cosine being the base, sine being the height.

Because the unit circle has the radius of one, all hypotenuse values will also be one, which is where the equation comes from

Graphical Explanation

**Problem Sets**

**Now that we have gone through the foundations of trigonometry, try these problems to test yourself!**

Radian Problems

Convert to Radian

120°

225°

330°

Convert to Degree

1.5 Radian

3π/2 Radian

19π/12 Radian

Unit Circle Problems

What does radian mean on a unit circle? How many radians are there in 2 full rotations of the unit circle?

What does cos135° mean on the unit circle?

True or False: Sin(π-x)=Sin(π+x). Justify your answer.

If Sin(x)= 2/5 and is in Quadrant II, find Cos(x). (Hint: Think back to Pythagorean identity)

Given a coordinate, (3π/4, √2/2) on y =sin(x), what does do the "x" and "y" values represent?

Solving Trigonometric Functions

Solve the following problems:

2 cos(x+1) -1

2 cos(x+3)= 0 (with the restriction of 0≤x≤2π

5sin²x - 4sinx-1=0 (with the restriction of 0≤x≤360°)

Do restrictions of a max of 360° and 2π mean the same thing?

Graphing and Transformations

What is the general equation for a sine graph? Is it the same of different for the cosine graph?

Why do cosine and sine wave differ? Do they differ at all?

Describe the amplitude and period for the equation:

y= -3 sin (1/2(x- π) -1/3

Graph y=1/2 cos (x-(π/6)) and indicate how it has changed from the parent graph. Also make a table of its x and y values.

Write a sine and cosine equation, in which the waves have a maximum of -4, minimum of -1, with a period of 2π, and that has shifted π/4 to the left.

How to Solve Trigonometric Equations

There are two main steps to keep in mind while solving trigonometric equations:

Step #1: Isolate the equation to cos(x) or sin(x)

Step #2: Locate all the places (there may be sometimes more than one) on the unit circle "x" will be appropriate for the equation

Side Notes:

* the period of the sin(x) and cos(x) function is 2π so, unless restricted, the "x" values of these equations will repeat every 2π radians in both directions, because sine and cosine waves are continual. So, to give a more exactly solution for all solutions, we would write the answer for x as added to 2πk, where k is an integer.

* consider the negatives and positives in the different quadrants of the unit circle

The Relationship Between Trigonometric Functions and Unit Circle

The answers to all trigonometric equations are given in radius form, that are all represented in the unit circle. When we look for "x" by evaluating sine and cosine on the unit circle, we are able to visualize the answer and understand what it really looks like. For example, if we are trying to solve for sin(x)= 1/2, I refer to the radian at which sin, or the height is equal to 1/2.

How does knowing the special case right triangle ratios help with finding radian without a calculator?

Technically, how many solutions could there be for an equation that equals any of the radians listed on the unit circle?

Trigonometric Function Problems

Formula for Finding the Radian

ALWAYS TRUE

SOMETIMES TRUE

NEVER TRUE

ONE CYCLE

ONE CYCLE

MULTIPLE CYCLES

In the purple triangle: Given that ø = 28°, x=3, find the missing variables

In the blue triangle, given A=130°, c=22, b=17, find all missing variables

In the multi-colored triangle, given that ø=40°, w=15, find all missing variables