Circling in on Angles (first chapter of Part 2)

Question 1

Name the 3 trig functions!

Answer 1

Sine, Cosine, Tagent

Question 2

Sine

1) How is it notated?
2) What is the ration?

Answer 2

1) sin θ
2)
opposite leg
_________
hypotenuse

Question 3

What are hypotenuse, opposide side, adjacent? Demonstrate by graphing the triangle!

Answer 3

hypotenuse = opposide from the right angle
opposide side = opposide from the angle which you search for
adjacent= what is left is adjacent leg

Question 4

How to find a 3rd side when you have given 2?

Answer 4

Use pythagoras. (leg)² + (leg)² = (hypotenuse)²
If you need a leg, put in leg 2 and hyp and solve for this leg. Keep in mind to take the SR because your variable is still squared.

Question 5

Cosine

1) How is it notated?
2) What is the ration?

Answer 5

1) cos θ
2)
adjacent leg
__________
hypotenuse

Question 6

1) Imagine having the cosine angle and the adjacent leg. You want to find out what the hypotenuse is. How to state the equation? What is X?

Answer 6

cos= adjacent leg
_________
x

Question 7

Tangent

1) How is it notated?
2) What is the ration?
3) What is another ratio of the tangent?

Answer 7

1) tan θ
2)
opposite leg 
\_\_\_\_\_\_\_\_\_
adjacent leg

3)
sin θ
____
cos θ

Question 8

Explain the reciprocal functions of of sine, cosine, and tangent! (Rarely used, so not too much memory place)

Answer 8

1) Cosecant, or csc θ, is the reciprocal of sine
- Thus, the ratio of hypotenuse to opposite leg

2) Secant, or sec θ, is the reciprocal of cosine.
- ratio of hypotenuse to adjacent leg.

3) Cotangent, or cot θ, is the reciprocal of tangent
- ratio of adjacent to opposite leg

Question 9

The inverse trig functions are arcsin, arccos, and arctan. In all these functions, theta (θ) is the input.

1) What are they useful for?
2) State all 3.

Answer 9

1) If you’re given the ratio of the sides and need to find an angle

2)  
Inverse sine (arcsin): θ = sin**-1(opp/hyp)
Inverse cosine (arccos): θ = cos**-1(adj/hyp)
Inverse tangent (arctan): θ = tan**-1(opp/adj)

Question 10

Where is the unit circle situated and what is his radius?

Answer 10

(0,0) with radius 1

Question 11

Finding out sin, cos, tan etc when you are given a (x,y) point in a coordinate system. Name the steps!

Answer 11

1. Draw a line from the point to the origin (0,0). This is your hypothenuse.
2. Draw a line from the point perpendicular to the x-axis. There is your right angle. Now you can just read the length of the two legs.
3. Use pythagoras to get the length of the hypothenuse (absolute values, no negative signs).
4. Plug in values in formulas depending on what you want.

Question 12

The 45°-45°-90° triangle
1) What are the lengths of the sides?

The 30°-60°-90° triangles
2) What are the lengths of the sides?

Answer 12

1) Both legs have the length a. Hypothenuse is a * SR2

2) short leg opposide from 30° = a. Longer leg= a*SR3. Hypothenuse= 2a

Question 13

The unit circle. Because the radius is 1, the hypothenuse is always 1 too. You start in quadrant 1, with 30,45,60 degrees … All values are positive. In the second quadrant(above left): x is positive and y negative and so on …

Answer 13

just information

Question 14

Find the trig functions of an angle (in this case, 225 degrees). Name the steps.

Answer 14

1)
- Draw it, to get a better feeling fo it. 225 is in the down left quadrant.
- To find out the degrees of the triangle: 225° is 45° more than 180°, so draw a 45°-
45°-90° triangle.

2)
- To find the lengths use the 45er triangle rules. Remember, hypothenuse is 1.
- Keep in mind, because the triangle is in the third quadrant, both the x
and y values should be negative.

3) Calculate for sine, cosine, tangent, cosecant, secant, contangent

Question 15

Radians show clear relationships between each of the
families on the unit circle and are used with reference angles.
θ’ (theta prime) is the name given to the reference angle, and θ is the actual solution to the equation. So you can find solutions by using the following quadrant rules: If …
QI: θ = θ’ because the closest way from the terminal side to the x-axis stays in the same quadrant. Reference angle and solution angle are the same…
what is it like for the other quadrants?

Answer 15

QII: θ = π – θ’ because it falls short of π by however
much the reference angle is.
QIII: θ = π + θ’ because you have π for 180 degrees, and the θ’ will go up the the left side of the X-axis.
QIV: θ = 2π – θ’ because it falls short of a full circle by
however much the reference angle is.

If you have questions, see screenshot 21.

Question 16

1) What is a reference angle?

2) What is the distance of a unit circle, if you go one time around? Not in degrees.

Answer 16

1) You have an angle. The angle on the other side, to the x-axis, is the reference angle. It is always under 90 degrees.
2) 2pi

Question 17

Solve 2 cos x = 1

Answer 17

1) Isolate for variable: cos x = 1/2

2) Determine which quadrants your solutions lie in.
- Cosine is also the x value(Look up ifyou don´t remember)
- Draw a triangle in ecery quadrant with x-axis legs labeled 1/2. Because it is non-negative, only Q1 and Q4 are possible.

3) Fill in the missing leg values for each triangle

4) Determine the reference angle.
- side length of 1/2 is the short leg of a 30°-60°-90° right
triangle. Therefore, the cosine (or the part along the x-axis) is the short leg and the vertical leg is the long leg. So the vertex of the angle at the center of the unit circle has a measure of 60°, making the reference angle π/3.

5) Express the solutions in standard form.
- The reference angle is θ’ = π/3. The first quadrant solution is the same as the reference angle: θ’ = π/3. The fourth quadrant solution is
θ = (2π) – θ’ = (2π) – π/3 = (5π)/33.

Question 18

What is π/6 in degrees?

Answer 18

π is 180 degree, so half a circle, which is 6π/6. So π/6 is 30 degree.

Question 19

How to calculate the length of an arc?

Answer 19

s= θ
__ * 2πr
360

2πr is the circumference of a circle
s represents the arc length
θ represents the measure of the angle in degrees

Because 360 is equal to 2π, what is left is s= θr, so angle times radius.