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 tips
5. Power Learn using a Mat Activity
6. Homework to reinforce and practice what you have learned.
1. 𝐹𝑖𝑛𝑑 sin 𝜃 , cos 𝜃 , 𝑎𝑛𝑑 tan 𝜃
B
6
Θ
A
C
4
2. Find all missing sides of the special right triangles.
1
A
30°
B
B
C
B
1
A
1
45°
C
A
3. Determine the reference angle and draw the reference triangle:
𝑎. 150°
𝑏. 225°
𝑐.
5𝜋
3
𝑑.
−5𝜋
4
60°
C
Click Here
for Solutions
1. 𝐹𝑖𝑛𝑑 sin 𝜃 , cos 𝜃 , 𝑎𝑛𝑑 tan 𝜃
B
6
A
Θ
4
C
36 = 16 + 𝑎2
20 = 𝑎2
2 5=a
2 5
5
=
6
3
4 2
cos 𝜃 = =
6 3
2 5
5
tan 𝜃 =
=
6
3
sin 𝜃 =
2. Find all missing sides of the special right triangles.
3. Determine the reference angle and draw the reference triangle:
5𝜋
−5𝜋
𝑎. 150°
𝑏. 225°
𝑐.
𝑑.
3
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……
A
Opposite
B
Θ
Adjacent
C
Next, we will learn three additional trigonometric ratios.
They are the reciprocals of sine, cosine, and tangent.
A
Opposite
B
Θ
Adjacent
C
Determine the values of the six trigonometric functions:
Determine
the ratios
and then
click here.
B
5
Using Pythagorean Theorem
we determine that AC = 4.
A
3
Θ
4
C
Determine the values of the six trigonometric functions in quadrant 1 when given:
B
7
A
Think of
the
solutions
then click
here.
Θ
2
C
Let’s consider our special right triangles and their trigonometric values:
1
A
30°
√𝟑
𝟐
B
B
√𝟐
𝟐
1
𝟏
𝟐
C
B
A
45°
√𝟐
𝟐
C
√𝟑
𝟐
1
A
60°
𝟏
𝟐
C
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.
Only sin and csc
are positive in
quadrant II
Only tan and cot
are positive in
quadrant III
S
A
T
C
All trig functions
are positive in
quadrant 1
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 the arrow if the
website worked.
Click this arrow if the
website did 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:
1
A
30°
√𝟑
𝟐
B
B
√𝟐
𝟐
1
𝟏
𝟐
C
B
A
45°
√𝟐
𝟐
1
60°
A 𝟏
𝟐
C
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.
30°
45°
60°
sinΘ
cosΘ
𝟏
𝟐
√𝟐
𝟐
√𝟑
𝟐
√𝟑
𝟐
√𝟐
𝟐
𝟏
𝟐
√𝟑
𝟐
C
Now, let’s
practice!
tanΘ
√3
1
√𝟑
𝟑
Let’s consider our special right triangles and their trigonometric values:
1
A
30°
√𝟑
𝟐
B
B
√𝟐
𝟐
1
𝟏
𝟐
C
B
A
45°
√𝟐
𝟐
C
√𝟑
𝟐
1
A
60°
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.
Now, you practice…
sinΘ
30°
45°
60°
cosΘ
C
Next Slide
or
Answers
tanΘ
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.
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. 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:
3𝜋
11𝜋
3𝜋
13. sec
1. sin 4
7. sin
2
6
5𝜋
4𝜋
14.
sin
−
8. sin 300°
2. cos
3
3
7𝜋
7𝜋
15.
cos
9. sec 120°
3. tan
4
6
4. cot −45°
10. sin 315°
5. sec −90°
11𝜋
11. cos
3
6. csc 390°
12. tan −
19. csc 𝜃 = 2,
cos 𝜃 < 0,
20. sec 𝜃 = 3 𝑎𝑛𝑑 𝑡𝑎𝑛𝜃 < 0,
5𝜋
4
𝑡ℎ𝑒𝑛 tan 𝜃 = _________
𝑡ℎ𝑒𝑛
sin 𝜃 = ________
19𝜋
16. cos −
6
17. tan
14𝜋
3
18. csc
17𝜋
6