Learning Goal: Students will then be able to use special right triangles to determine geometrically the sine, cosine and tangent for angles that

Agenda:

1. Prior Knowledge Check: Pythagorean Theorem, Definition of Sine, Cosine, and Tangent, Special Right Triangles, Reference Angle and Triangle in degrees or radians.

2. Lesson and Guided Practice: Six Trig Functions, Trig Functions defined in terms of the unit circle

3. Practice Exact Trig Values via the website: http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php

4. Organize and Synthesize exact trig values using a table and then using finger tips5. Power Learn using a Mat Activity 6. Homework to reinforce and practice what you have learned.

ΘA

B

C

1.𝐹𝑖𝑛𝑑sin 𝜃 ,cos𝜃 ,𝑎𝑛𝑑 tan𝜃6

4

A

B

C A

B

C A

B

C30 ° 45 ° 6 0°

2. Find all missing sides of the special right triangles.

1 1 1

3. Determine the reference angle and draw the reference triangle:

𝑎 .150 °𝑏 .225 °𝑐 . 5𝜋3 𝑑 . −5𝜋4

Click Here for Solutions

ΘA

B

C

1.𝐹𝑖𝑛𝑑sin 𝜃 ,cos𝜃 ,𝑎𝑛𝑑 tan𝜃6

4

2. Find all missing sides of the special right triangles.

3. Determine the reference angle and draw the reference triangle:

𝑎 .150 °𝑏 .225 °𝑐 . 5𝜋3 𝑑 . −5𝜋4

20 = 2= a

In Geometry we learned three trigonometric ratios made from the sides of a right triangle. They are sine, cosine, and tangent.

We had an acronym to remember the ratios……SohCahToa……

ΘA

B

C

Hypotenuse

Opp

osite

Adjacent

Next, we will learn three additional trigonometric ratios. They are the reciprocals of sine, cosine, and tangent.

ΘA

B

C

Hypotenuse

Opp

osite

Adjacent

ΘA

B

C

Determine the values of the six trigonometric functions:

53

Using Pythagorean Theorem we determine that AC = 4.

4

Determine the ratios and then

click here.

ΘA

B

C

Determine the values of the six trigonometric functions in quadrant 1 when given:

72

Think of the

solutions then click

here.

Let’s consider our special right triangles and their trigonometric values:

A

B

C A

B

C A

B

C30 ° 45 ° 6 0°

1 1 1𝟏𝟐

√𝟑𝟐

√𝟐𝟐

√𝟐𝟐

𝟏𝟐

√𝟑𝟐

If we let the hypotenuse be 1, then sinΘ=y and cosΘ=x . Since the hypotenuse is the radius of a unit circle, we consider it to be radius = r = 1.

Practice the six trig functions for each of the

special angles looking at the triangles above.

You can draw the appropriate triangle, with the reference angle in standard position, having the radius = 1. Next, using your special right triangles skills, you can determine the three basic trig functions: sin, cos, and tan, and then, their reciprocals.

If you know these trig functions, you will be able to determine the values in other quadrants!

Name the six trig functions for each of the three reference triangles above.

Let this be a circle with radius one. Do you see the angles with a 30° reference angle are indicated with a green dot, angles with a 45° reference angle are indicated with a red dot, angles with a 60° reference angle are indicated with a blue dot, and quadrantals are indicated with a pink dot.

Drawing the reference triangle in any of the four quadrants, will give the same numerical answer for the trig functions, however the sign of the trig functions may be different. We can use the saying, “All Students Take Calculus” to help remember the signs.

S A

CT

All trig functions are positive in

quadrant 1

Only sin and csc are positive in

quadrant II

Only tan and cot are positive in

quadrant III

Only cos and sec are positive in

quadrant IV

Now, let’s investigate the “quadrantals”.

When the terminal side of an angle Θ that is in standard position lies on one of the coordinate axes, the angle is called a quadrantal angle. The terminal sides of these angles would be located at the pink dots.

Since the coordinates of the pink dots will have 0 and 1, we need to remember the division rules with 0.

http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php

Let’s practice…..When you click the link below, adjust your screen so that you see the circle. Practice the capabilities of the website, by following the directions and doing some trial and error. When finished practicing, deselect the little boxes and choose “quiz me”. Practice determining sine and cosine, before clicking the flashing circle. Be sure to select other quadrants as well as the first quadrant.

Click the arrow if the website worked.

Click this arrow if the website did NOT work.

http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php

These circles can be used if the website does not work. Select a dot for the terminal side, then determine since and cosine of the reference triangle.

Click the arrow to check your answers.

Click the arrow after sufficient practice to continue.

http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php

These circles can be used if the website does not work.

Click the arrow to return to the uncovered circle.

Skip practice slide

Let’s consider our special right triangles and their trigonometric values:

A

B

C A

B

C A

B

C30 ° 45 ° 6 0°

1 1 1𝟏𝟐

√𝟑𝟐

√𝟐𝟐

√𝟐𝟐

𝟏𝟐

√𝟑𝟐

If we let the hypotenuse be 1, then sinΘ=y and cosΘ=x . Since the hypotenuse is the radius of a unit circle, we consider it to be radius = r = 1.

sinΘ cosΘ tanΘ

30° √3

45° 1

60°

𝟏𝟐

√𝟑𝟐

√𝟑𝟐

𝟏𝟐

√𝟐𝟐

√𝟐𝟐

√𝟑𝟑

Now, let’s practice!

Let’s consider our special right triangles and their trigonometric values:

A

B

C A

B

C A

B

C30 ° 45 ° 6 0°

1 1 1𝟏𝟐

√𝟑𝟐

√𝟐𝟐

√𝟐𝟐

𝟏𝟐

√𝟑𝟐

If we let the hypotenuse be 1, then sinΘ=y and cosΘ=x . Since the hypotenuse is the radius of a unit circle, we consider it to be radius = r = 1.

sinΘ cosΘ tanΘ

30°

45°

60°

Now, you practice… Answers

Next Slideor

Your hand can help you remember…..

Use the following finger tricks!

Hold your hand to remind you of the special angles in the first quadrant.

Think of cosine on top and sine on the bottom. Consider that for the three

specials angles, every answer will be a radical value over two.

Here we go!Fold your angle finger back and

fill-in the radicand with the number of

fingers:top for cosine

and bottom for sine.

sin 30 °=√𝟏2

=12

cos 30°=

√𝟑2

sin 45 °=√𝟐2

cos 45 °=√𝟐2

sin 60 °=√𝟑2

cos 60 °=√𝟏2

The basic trig functions that we have practiced will be used throughout all of PreCalculus and also in Calculus. Therefore, it is important to know them as much as you know your multiplication facts and other computations in mathematics.

Time yourself to complete the “Mat Activity”. Can you complete your “Mat” in under 2 minutes? This knowledge will serve you well……

The mat is a set of trig functions to be placed in a sheet protector.The solution square are cut-out squares to be placed randomly, face up around the mat. The teacher will use a timer to have students begin and put the appropriate answer square next to its problem. This should be completed in under two minutes….

or try again!

Play count down PowerPointto time the Mat Activity.

The next slide has the homework. It should be started in class and finished at home.

Upon completion, feel free to check your answers using the website:

http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php

Practice: Sketch the reference triangle and determine the exact value of each expression:

1.

2. cos 4 𝜋3

3. tan 7𝜋6

4. cot (−45 ° )

5. sec (−90 ° )

6.csc (390 ° )

7. sin 11𝜋6

8. sin (300 ° )

9.

10. sin (315 ° )

11.cos 11𝜋312.

13. sec 3𝜋2

14. sin(− 5𝜋3 )15. cos 7𝜋4

16. cos (− 19𝜋6 )17. tan ( 14𝜋3 )18. csc( 17𝜋6 )

19.csc𝜃=2 ,cos𝜃<0 , h𝑡 𝑒𝑛 tan𝜃=¿¿¿¿

20. sec 𝜃=√3𝑎𝑛𝑑𝑡𝑎𝑛𝜃<0 , h𝑡 𝑒𝑛 sin𝜃=¿¿¿