Trig Unit Unit Circle Lesson

The Unit Circle

Outcome: Use the unit circle to find arc lengths on a circle and evaluate trigonometric functions

​What is a Quadrant?

Each of the 4 quarters of a circle

• Quadrants of a circle are numbered

  going counter clockwise​

• The quadrant determines whether

  the x and y values are positive

  or negative​

Radians vs Degrees

Multiple Choice

How many degrees are in one revolution?

• $$90\degree$$

• $$360\degree$$

• 180

Multiple Choice

How many radians are in one revolution?

• $$\pi$$

• $$\frac{\pi}{2}$$

• $$2\pi$$

Converting Degrees into radians

Multiple Choice

How many radians are in $$60\degree$$ ?

• $$\frac{\pi}{2}$$  

• $$\frac{3\pi}{2}$$  

• $$\frac{\pi}{3}$$  

Converting Radians to degrees

Multiple Choice

How many degrees are in $$\frac{\pi}{4}$$  radians?

• $$30\degree$$  

• $$45\degree$$  

• $$60\degree$$  

Multiple Choice

How many degrees are in $$\frac{2\pi}{3}$$  radians?

• $$90\degree$$  

• $$120\degree$$  

• $$150\degree$$  

The triangle at the right is a right triangle, and θ is in standard position.

Fill in the Blanks

What is the length of the hypotenuse of the triangle? (Think about it as a radius)

OP=

Label OP on your paper.

Match

Using θ as a reference angle, mark the triangle sides as Opp, Adj, and Hyp.

Label these on the triangle on your paper.

• x

• y

• OP

Adj

Opp

Hyp

• Adj

• Opp

• Hyp

Match

Now, use the given triangle to represent $$\sin\theta$$ and $$\cos\theta$$ in terms of the variables given in the diagram.

• $$\frac{y}{1}=y$$

• $$\frac{x}{1}=x$$

• 1

sinθ

cosθ

OP

• sinθ

• cosθ

• OP

Drag and Drop

What about tangent? Use the triangle to answer the following questions.



In terms of x and y, what does tanθ equal?

tanθ=​ ​

Placeholder

In terms of sinθ and cosθ, what does tanθ equal?

tanθ=​

Placeholder

Drag these tiles and drop them in the correct blank above

x

y

sinθ

cosθ

Using the Unit Circle

• to evaluate sine and cosine, we use the coordinates that lie on the Unit Circle's circumference

• cosine of an angle is equal to the length of the reference triangle's adjacent side and is represented by the x-coordinate

• sine of an angle is equal to the length of the reference triangle's opposite side and is represented by the y-coordinate

REMEMBER

Evaluate Sine

• Sin(angle) = y-coordinate of point

• Sine is positive in the FIRST and SECOND quadrants.

• Sine is negative in the THIRD and FOURTH quadrants.

• sin(135°) = √2/2

• sin(4π/3) = -√3/2

• sin(0) = 0

Multiple Choice

Evaluate  $$\sin\left(\frac{\pi}{2}\right)$$  

• $$-\frac{1}{2}$$

• $$\frac{1}{2}$$

• $$\frac{\sqrt{3}}{2}$$

• $$-1$$

• $$1$$

Evaluate Cosine

• Cos(angle) = x-coordinate of point

• Cosine is positive in the FIRST and FOURTH quadrants.

• Cosine is negative in the SECOND and THIRD quadrants.

• cos(2π/3) = -1/2

• cos(270°) = 0

• cos(π/6) = √3/2

Multiple Choice

Evaluate cos(225°)

• $$-\frac{\sqrt{2}}{2}$$

• $$\frac{\sqrt{2}}{2}$$

• $$-\frac{\sqrt{3}}{2}$$

• $$-\frac{1}{2}$$

Evaluate Tangent

• Tan(angle) = y-coordinate/x-coordinate

• Tangent is positive in the FIRST and THIRD quadrants.

• Tangent is negative in the SECOND and FOURTH quadrants.

• tan(90°) = 1/0 = UNDEFINED

• tan(π) = 0/-1 = 0

• tan(240°) = (-√3/2)/(-1/2)

• =(-√3/2) *(-2/1) = √3/1 = √3

Multiple Choice

Evaluate $$\tan\left(\frac{7\pi}{6}\right)$$  

• $$-\frac{\sqrt{3}}{2}$$  

• $$\frac{\sqrt{3}}{3}$$  
  

• $$-\frac{1}{2}$$  

• $$\sqrt{3}$$