Agenda: Learning Goal: Students will then be able to use special right triangles to determine geometrically the sine, cosine and tangent for angles that Agenda: Prior Knowledge Check: Pythagorean Theorem, Definition of Sine, Cosine, and Tangent, Special Right Triangles, Reference Angle and Triangle in degrees or radians. Lesson and Guided Practice: Six Trig Functions, Trig Functions defined in terms of the unit circle Practice Exact Trig Values via the website: http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php Organize and Synthesize exact trig values using a table and then using finger tips Power Learn using a Mat Activity Homework to reinforce and practice what you have learned.

2. Find all missing sides of the special right triangles. 1. 𝐹𝑖𝑛𝑑 sin 𝜃 , cos 𝜃 , 𝑎𝑛𝑑 tan 𝜃 B 6 Θ C A 4 2. Find all missing sides of the special right triangles. A B C A B C A B C 1 1 1 30° 45° 60° 3. Determine the reference angle and draw the reference triangle: 𝑎. 150° 𝑏. 225° 𝑐. 5𝜋 3 𝑑. −5𝜋 4

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1. 𝐹𝑖𝑛𝑑 sin 𝜃 , cos 𝜃 , 𝑎𝑛𝑑 tan 𝜃 sin 𝜃= 2 5 6 = 5 3 cos 𝜃= 4 6 = 2 3 tan 𝜃= 2 5 6 = 5 3 B 36=16+ 𝑎 2 20 = 𝑎 2 2 5 = a 6 Θ C A 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. 5𝜋 3 𝑑. −5𝜋 4

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…… Θ A B C Hypotenuse Opposite Adjacent

Next, we will learn three additional trigonometric ratios Next, we will learn three additional trigonometric ratios. They are the reciprocals of sine, cosine, and tangent. Θ A B C Hypotenuse Opposite Adjacent

Determine the ratios and then click here. Determine the values of the six trigonometric functions: Determine the ratios and then click here. Θ A B C 5 Using Pythagorean Theorem we determine that AC = 4. 3 4

Think of the solutions then click here. Determine the values of the six trigonometric functions in quadrant 1 when given: Θ A B C 7 2 Think of the solutions then click here.

Let’s consider our special right triangles and their trigonometric values: B C A B C A B C √𝟑 𝟐 1 1 √𝟐 𝟐 1 𝟏 𝟐 30° 45° 60° √𝟑 𝟐 √𝟐 𝟐 𝟏 𝟐 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 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. Only sin and csc are positive in quadrant II All trig functions are positive in quadrant 1 S A Only tan and cot are positive in quadrant III T C Only cos and sec are positive in quadrant IV 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.

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.

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. http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php Click this arrow if the website did NOT work. Click the arrow if the website worked.

These circles can be used if the website does not work 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. http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php

Skip practice slide Let’s consider our special right triangles and their trigonometric values: A B C A B C A B C √𝟑 𝟐 1 1 √𝟐 𝟐 1 𝟏 𝟐 30° 45° 60° √𝟑 𝟐 √𝟐 𝟐 𝟏 𝟐 Now, let’s practice! 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° √𝟑 𝟐 𝟏 𝟐 √𝟐 𝟐 √𝟐 𝟐 √𝟑 𝟐 √𝟑 𝟑 𝟏 𝟐

Let’s consider our special right triangles and their trigonometric values: B C A B C A B C √𝟑 𝟐 1 1 √𝟐 𝟐 1 𝟏 𝟐 30° 45° 60° √𝟑 𝟐 √𝟐 𝟐 𝟏 𝟐 Next Slide 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. or Answers Now, you practice… sinΘ cosΘ tanΘ 30° 45° 60°

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.

top for cosine and bottom for sine. 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. cos 60°= 𝟏 2 sin 60°= 𝟑 2 cos 30°= 𝟑 2 cos 45°= 𝟐 2 sin 30°= 𝟏 2 = 1 2 sin 45°= 𝟐 2

The mat is a set of trig functions to be placed in a sheet protector. 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 PowerPoint to time the Mat Activity.

The next slide has the homework 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: 13. sec 3𝜋 2 7. sin 11𝜋 6 1. sin 3𝜋 4 14. sin − 5𝜋 3 2. cos 4𝜋 3 8. sin 300° 15. cos 7𝜋 4 3. tan 7𝜋 6 9. sec 120° 16. cos − 19𝜋 6 10. sin 315° 4. cot −45° 11. cos 11𝜋 3 17. tan 14𝜋 3 5. sec −90° 12. tan− 5𝜋 4 6. csc 390° 18. csc 17𝜋 6 19. csc 𝜃=2, cos 𝜃<0, 𝑡ℎ𝑒𝑛 tan 𝜃=_________ 20. sec 𝜃= 3 𝑎𝑛𝑑 𝑡𝑎𝑛𝜃<0, 𝑡ℎ𝑒𝑛 sin 𝜃=________