Review: Right Triangles and Trigonometry

Trigonometry use after wednesday.notebookOctober 22, 2020
Review: Right Triangles and Trigonometry Create a list of everything you know about a right triangle.
Trigonometry use after wednesday.notebook
October 22, 2020
How do you find the third side of a triangle if given two sides?
Example: Find the length of the hypotenuse of a right triangle, in simplest radical form, if the legs of the triangle have a length of 2 and 4.
October 22, 2020
How do you find the third side of a triangle if given one side and one angle?
Example: Find the length of the hypotenuse of a right triangle if the leg opposite its 30 degree angle has a length of 1.
What do you think of when you hear this topic?
October 22, 2020
Find the missing side of each in simplest radical form where appropriate.
October 22, 2020
Find the missing side of each in simplest radical form where appropriate.
October 22, 2020
Review Do Now: Draw and label the sides and angles of the two special right triangles.
October 22, 2020
Why are the special right triangles helpful? Evaluate sin(60) using your calculator.
Using your special right triangle, evaluate sin(60) without using your calculator.
October 22, 2020
The special right triangles allow us to have common trig values.
Fill in the table of common trig values below:
October 22, 2020
Using your table or special right triangles only (no calculator), evaluate the following:
What are our three basic trig functions?
These trig functions also have reciprocals and their own special names.
October 22, 2020
Using your table or special right triangles only (no calculator), evaluate the following:
October 22, 2020
Homework Page 335 #1 Find the values of all six trig functions of the angle theta.
#9 (hint: draw a triangle and fill in the sides based on SOH CAH TOA) Evaluate the remaining trig functions given
#26 (find without a calculator)
#49 (using a calculator) Solve for the variable shown.
45
3
October 22, 2020
Are triangles the only shape that has angles measured in degrees?
Name a shape that has interior angles that can be to more than 180 degrees.....more than 360 degrees?
October 22, 2020
In a triangle you are limited since the sum of the angles is 180 degrees.
With a circle you can wrap around the circle as many times as you want creating larger and larger angles.
Facts about the unit circle:
Center = ( , )
Radius = ______
x = ________
y = ________
October 22, 2020
In which quadrants of the unit circle are the sine, cosine and tangent positive?
October 22, 2020
Name the quadrant in which the angle, θ, must lie when:
1) cos θ > 0 and sin θ > 0
2) cos θ < 0 and sin θ < 0
3) cos θ > 0 and sin θ < 0
4) cos θ > 0 and tan θ > 0
5) sin θ < 0 and tan θ < 0
6) csc θ > 0 and sec θ > 0
7) cot θ > 0 and sec θ < 0
8) csc θ < 0 and cot θ < 0
October 22, 2020
Now that we have access to the unit circle, we are able to work with more angles!
The angles that lie on the axes are known as quadrantal angles.
Label the 4 quadrantal angles.
Since this is the unit circle, label the coordinates where the circle intersects the axes.
October 22, 2020
Using the labeled Unit Circle, find the trig values for the following angles:
October 22, 2020
When given a positive angle, how do you think we can find its coterminal angle?
When given a negative angle, how do you think we can find its coterminal angle?
These larger angles will have the same points though as their smaller angles, these pairs are known as _______________.
11) cos θ = 5/11
25) Given an exact value sec 45
#4144 (find the angle in degrees, use your special right triangles and not your calculator)
41) sin θ = 1/2
Page 347 #1
1) Identify the one angle that is not coterminal with all the others.
150, 510, 210, 450, 870
October 22, 2020
Name the quadrant in which the angle lies and draw a picture of the rotation.
October 22, 2020
For each given angle, find the angle of smallest positve measure which is coterminal with the given angle.
October 22, 2020
Scenario: Over quarantine I picked up a new crafting hobby, making jewelry pendants. I like to divide up the circular pendants and put different fabrics and materials along the edge. I ran into a slight problem.
When I divided up the sections of the circle it was measured in degress and I couldn't ask for material at store in degrees, they needed a length measure.
October 22, 2020
The biggest reason why we needed to learn radians is that we are in College Pre-Calculus now, higher level math.
All higher level math theorems starting in the course and beyond are based upon radian measure, working in degrees unravels math from now on.
When given degrees, you must convert to radian measure!
October 22, 2020
Homework Page 325 #9, 11, 17, 21, 25, 27, 29, 36
#9, 11 Convert from degree to radian
9) 60
11) 120
17)
21)
25) s = ?, r = 2 in., θ = 25 radians
27) s = 1.5 ft, r = ?, θ = radians
29) s = 3 m, r = 1 m, θ =?
36) A 100 degree arc of a circle has a length of 7 cm. To the nearest centimeter, what is the radius of the circle?
October 22, 2020
Take out your unit circle worksheet. Now that we have the quadrantal angles filled in, let's label some other important angles in radian measure.
October 22, 2020
Is there a better way to label the angles in radian? Will we always have to convert?
In your notebook, draw a unit circle.
What happens if we count by ? ?
Example 1: Construct an angle of radians in standard position.
Coterminal Angles:
Example: Find angles in standard position that are coterminal with an angle of:
(a) (b)
Page 347 Quick Review 4.3 (at the top) #15,7
Give the value in degrees
1)
2)
3)
4)
5)
7)
Page 347 Section 4.3 Exercises #2
2) Identify the one angle that is not coterminal with all the others.
Go back to finish coterminal angles
October 22, 2020
Draw a unit circle on a piece of paper. Then without looking at our notes from yesterday fill in as many angles as you can in radian measure.
There are 16 in total that should be labeled.
October 22, 2020
Do Now: In the beginning we spoke about the 306090 and 454590 special right triangles. This was bad. We are no longer working with degrees, please convert those to radian measure. Then complete (1) and (2). 1) Find all sides of a 306090 triangle with a hypotenuse of 1.
2) Find all sides of a 454590 triangle with a hypotenuse of 1.
October 22, 2020
Using the special right triangles, find the exact value (without a calculator) of:
a)
b)
c)
d)
e)
f)
g)
h)
October 22, 2020
1. Carefully fold your plate in half. Then take your semicircle and fold it again to make quarter circles.
2. Unfold your plate and color in both creases (in pencil or black pen), using a straight edge. These are your axes label them +x, x, +y, y.
Check in: Where are the four quadrants? Where is x positive? Where is y positive?
3. Now fold the bottom of the plate up (along the xaxis) and then fold the left side onto the right side (along the yaxis). The "corner" should be on the bottom lefthand side.
4. Let's make some marks, in different colors, on the folded plate:
Count 9 ridges (1/2 of the way) and make a blue mark on the inside & outside of the plate at that point, so that it marks all four layers.
Count 6 ridges (1/3 of the way) from the bottom and make a red mark, again so that it marks all four layers.
Count 12 ridges (2/3 of the way) from the bottom and make a green mark, again so that it marks all four layers.
5. Open up your plate. You should see 4 red marks, 4 blue marks, and 4 green marks. Using the color of the marks, draw a straight line from the origin (center) to the tick marks ONLY in the FIRST QUADRANT.
1
2
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6. Now let's see how this all relates to what we already know. Fill out side 1 of the worksheet.
7. Write those coordinates on your unit circle plate!
October 22, 2020
8. Can you now figure out the coordinates of the other 9 points on your plate? Write these in the appropriate spots on the second page of the worksheet.
9. If each of these lines represent terminal sides of angles in standard position... can you figure out each of these angles? Write the measure of each angle on its terminal side. Start with the quadrantal angles.
October 22, 2020
So.... how does this help us figure out our original questions?
Without a calculator, evaluate:
a. Evaluate:
b. Evaluate:
c. Evaluate:
d. In one counterclockwise rotation, for what angles θ is sin(θ) positive?
e. In one counterclockwise rotation, for what angles θ is cos(θ) positive?
f. Find four different angles that satisfy the equation sin(θ)=1/2
g. Find all solutions to the equation sin(θ)=1/2
Are sin(θ), cos(θ), and tan(θ) the only three trigonometric functions?
October 22, 2020
Using your plate, for 0≤θ≤2π, solve each equation for θ.
1) sin θ = 1/2
2) cos θ = 0
October 22, 2020
Which quadrant do you think is the most important to know from our plate?
How can we find the other quadrants from the
first quadrant?
October 22, 2020
Do Now: Recreate the first quadrant with the angles and points.
Evaluate
5)
6)
7)
8)
27)
28)
29)
30)
31)
32)
October 22, 2020
Without using your plate, for 0≤θ≤2π, solve each equation for θ.
1) sin θ = 1/2 4) tanθ = 1
2) sin θ = 1/2 5) cosθ = 1
3) cos θ = 0 6) cotθ = 0
Do Now: What should you study for the assessment tomorrow?
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Review: Right Triangles and Trigonometry

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