Staying WarmSuppose you are given a ray

on the coordinate plane andits endpoint P.

How would you find the length of the ray r?

Assume 𝑟 ≠ 0.

Find all six trigonometric ratios in terms of x, y, and r.

Unit 4.3: Trigonometric Functions and the Unit Circle

Straight to the point

Find the 6 trig functions for the following points

1. (8, -6) 2. (4,3) 3. (-2, -1)

For Future Reference

If 𝜃 is an angle in standard position, its reference angle 𝜃′ is the acute angle formed by the terminal side of 𝜃 and the x-axis.

Find the reference angle for the following angles.

1. 300𝑜 2. −2𝜋

33.

5𝜋

4

4. 240𝑜 5. 390𝑜

Using the trig ratios from the warm-up, you come up with the following ratios based on quadrant:

Using the trig ratios from the warm-up, you come up with the following ratios based on quadrant:

Today’s Special

Recall the side lengths for the two special right triangles:

If the length of the hypotenuse in each triangle is 1, what is the length of the legs of the triangles?

You’re the Investigator

On the unit circle, the rays will always have a length of 1 unit (hence the name).

Using your knowledge of the special right triangles, you will be able to find the points on the unit circle for all of the multiples of 30𝑜 and 60𝑜

Reference angles will be useful in completing the unit circle.

After you have found the points on the unit circle, you can then find 𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃, 𝑎𝑛𝑑 𝑡𝑎𝑛𝜃 for all multiples of 30𝑜 and 45𝑜.

Note that you will always let 𝑐𝑜𝑠𝜃 = _____and 𝑠𝑖𝑛𝜃 =_____in your equation.

This leads to 𝑡𝑎𝑛𝜃 = _______

Being really familiar with the unit circle is really important.

It’s pop-quiz important.

It’s more-than-one-pop-quiz-a-week important.

It’s pop-quiz-without-your-notes important.

Use your completed unit circle to findwhen each of the basic trig functionsare positive and whenthey are negative.

This leads to the acronym that helps us remember when each trig function ispositive or negative.

Remember, Remember

Appalachian State Teaching College

All Students Take Calculus

All Students Take Chemistry

All Silly Tom Cats

Add Sugar To Coffee

Or make up your own mnemonic

Find the exact value of each expression.

1. sin(𝜋

3) 2. 𝑐𝑜𝑠135𝑜

3. 𝑡𝑎𝑛270𝑜 4. csc(11𝜋

6)

5. cos(𝜋

4) 6. 𝑠𝑖𝑛120𝑜

7. 𝑐𝑜𝑡210𝑜 8. sec(7𝜋

4)

Periodic functions are functions with values that repeat at regular intervals.

You can use this periodic nature of trigonometric functions to find trig values for angles that don’t fall between 0 − 360𝑜 using coterminalangles.

Find the exact value of each expression.

1. cos(11𝜋

4) 2. sin(−

2𝜋

3)

3. tan(19𝜋

6) 4. sin(

13𝜋

4)

5. cos(−4𝜋

3) 6. tan(

15𝜋

6)

Buy One, Get 5

When given a trig ratio and the sign of a second ratio, find the remaining 5 trig ratios.

1. 𝑡𝑎𝑛𝜃 = 5/12 and 𝑠𝑖𝑛𝜃 < 0

2. 𝑠𝑒𝑐𝜃 = 3 and 𝑡𝑎𝑛𝜃 < 0

3. 𝑠𝑖𝑛𝜃 = 5/7 and 𝑐𝑜𝑡𝜃 > 0

Domo Arigato

As part of the range of motion category in a high school robotics competition, a student programmed a 20-cm long robotic arm to pick up an object at point C and rotate through an angle of exactly 225𝑜 in order to release it into a container at point D. Find the position of the object at point D, relative to the pivot point O.

Through Angular Velocity: p238 #34-40 Even

Through 6 Trig Functions: p251 #2-8 Even

Through Reference Angle: p251 #18-24 Even

Through Unit Circle: p251 #10-16 Even and #26-32 Even

Through Periodic Nature: p251 #44-50 Even

Through B1G5: p251 #34-40 Even

Through Mr. Roboto: p251 #41, 42