Topic 6.1-6.3 Quiz

Example of a Reference Angle

Reference angle of 130 is 50 180 - 130 === 50 Reference angle of 210 is 30 210 - 180 === 30

What is the distance the satellite travels while it is being tracked?

The total fistance from the center of Earth to the satellite is 6,400 km + 320 km = 6720 km Earth's radius - 6,400 km Above Earth's surface - 320 km

Fundamental Pythagorean Identity

(sin x)^2 + (cos x)^2 = 1 Related Identities Divide both sides of the above by cos^2x and simplify (sin^2 (x) / cos^2 (x)) + (cos^2 (x) / cos^2 (x)) = 1/cos^2 (x) tan^2(x) + 1 = sec^2 (x)

What is the sec, csc, and cot of -10π/4?

-10π/4 = -5π/2 = -450 = +270 sec 270 = 1/cos 270 = 1/0 === 0 csc 270 = 1/sin 270 = 1/-1 === -1 cot 270 = cos 270/sin 270 = 0/-1 === 0

UNIT CIRCLE - NEED TO KNOW

0 degrees = 0 30 degrees = π/6 45 degrees = π/4 60 degrees = π/3 90 degrees = π/2 120 degrees = 2π/2 135 degrees = 3π/4 150 degrees = 5π/6 180 degrees = π 210 degrees = 7π/6 225 degrees = 5π/4 240 degrees = 4π/3 270 degrees = 3π/2 300 degrees = 5π/3 315 degrees = 7π/4 330 degrees = 11π/6 360 degrees = 2π

Example Problems with the Unit Circle

1. cos π/3 = 60 === 1/2 2. sin 5π/3 = Q4 - sin (360-300) = - sin 60 = -√3/2 3. tan -2π/3 (-120)(+240) Q3 + tan (240 - 180) tan 60 = √3 4. sec -π/4 sec - 45 sec 315 1/cos 315 - 1/(2/√2) === √2 cos 315 Q4 cos 360-315 cos 45 = √2/2 5. csc 3π/4 csc 315 Q2 - 1/sin 135 sin 135 = sin (180-135) = sin 45 = √2/2 = 1 / √2/2 === √2 6. cot 7π/4 = 315 1/tan 315 Q4 tan 315 = - tan (360-315) = - tan 45 === -1

What are the sine and cosine of the angle 2π/3?

Angle - 120 Reference angle - 60 sin 60 = √3/2 cos 60 === -1/2

What is x, y in this situation? Triangle is a 45-45-90, Hypotenuse is 1

Both legs will be √2/2 as this is the rule in a 45-45-90

Converting from Degrees to Radians and Radians to Degrees

Degrees --> Radians Degrees * π/180 Ex. 30 degrees = 30 * π/180 = 30π/180 = π/6 radians Radians --> Degrees Radians * 180/π Ex. π/6 = π/6 * 180/π = 180/6 = 30 degrees

Given the initial and terminal sides, find a positive angle measure, a negative angle measure, and an angle measure greaer than 360 for each angle below

Half circle from Q1 to Q2 - terminal side in Q2 at 180 +180 degrees - positive \ -180 degrees - negative > Co-terminal Angles 540 degrees - >360 / Angle going into Q3 - 65 degrees after 180 +245 degrees - positive \ -115 degrees - negative > Coterminal Angles 605 degrees - >360 /

What are the locations of opposite, adjacent, and hypotenuse?

If the hypotenuse is on the right side, the bottom length is the opposite, and the length directly across from the hypotenuse is the adjacent.

Side Lengths of a 30-60-90 Triangle

Length of sides: x - adjacent x √3 - opposite 2x - hypotenuse

Side Lengths of a 45-45-90 Triangle

Lengths of sides x - adjacent x - opposite x√2 - hypotenuse

Rules of Each Quadrant in the Unit Circle

Q1 - Leave alone, ALL POSITIVE Q2 - 180 - angle, SIN and CSC POSITIVE Q3 - angle - 180, TAN and COT POSITIVE Q4 - 360 - angle, COS and CSC POSITIVE | 0 | 90 | 180 | 270 | 360 | -------------------------------------------------------------------- sin | 0 | 1 | 0 | -1 | 0 | -------------------------------------------------------------------- cos | 1 | 0 | -1 | 0 | 1 | -------------------------------------------------------------------- tan | 0 | U | 0 | U | 0 | --------------------------------------------------------------------

Rules for Each Quadrant with Reference Angles

Q1 --> 0 < θ < 90 -360 < θ < -270 Q2 --> 90 < θ < 180 -270 < θ < -180 Q3 --> 180 < θ < 270 -180 < θ < -90 Q4 --> 270 < θ < 360 -90 < θ < 0

What is cos θ if sin θ = -0.8 and the angle with the measure θ is in quadrant 4?

SOH-CAH-TOA -0.8 = -4/5 Opposite is -4 Hypotenuse is 5 Opposite must be 3 Quadrant 4 so it is positive cos θ === 3/5

What is sin θ if cos θ = -3/5 and the angle with the measure θ is in quadrant 3?

SOH-CAH-TOA Adjacent is -3 Hypotenuse is 5 Opposite must be 4 Quadrant 3 so it is negative sin θ === -4/5

Example of Reference Angles with Radians

Suppose an angle measures 4π/3 radians. Its intercepted angle on the unit circle is also 4π/3 radians The top half of the circle represents π radians. 4π/3 is greater than π radians by less than 3π/2 radians. Therefore, an angle with the measure of 4π/3 radians lies in Q3 and has a reference angle of π/3

Rules with Radian Measure of a Central Angle

The radian measure of a central angle is equal to the length it is on the unit circle subtended by that angle. An angle measure of 1 radian subtends and are on the unit circle with length 1.

Rules of Trigonometric Ratios

The three sides of a right triangle are referred to as the hypotenuse and two legs. The greek letter θ, read theta, is often used to represent an acute angle in a right triangle. Angle θ is an abbreviation for, "Angle with measure θ."

What is the sec, csc, and cot of 210?

sec 210 = sec 30 -1/cos 30 = 1/(√3/2) === -2√3/3 csc 30 = 1/sin 30 = 1/(1/2) === -2 cot 30 = -1/tan 30 = 1/(√3/3) === -√3

Example of the ratios for the six base trigonometric functions given a triangle with leg lengths of 12 and 5, and a hypotenuse length of 13

sin = 5/13 cos = 12/13 tan = 5/12 csc = 13/5 sec = 13/12 cot = 12/5

Triangle MNO is a 45 - 45 - 90 triangle with side length OM 2. What are the six trigonometric ratios for angle N with a measure θ?

sin N = 2/2√2 = 1/√2 * √2/√2 === √2/2 cos N = 2/2√2 = 1/√2 * √2/√2 === √2/2 tan N = 2/2 = 1 csc N = 2√2/2 = √2 sec N = 2√2/2 = √2 cot N = 2/2 = 1

What are the six base trigonometric functions of the angle θ?

soh - cah - toa sine - sin θ = opposite/hypotenuse cosine - cos θ = adjacent/hypotenuse tangent - tan θ = opposite/adjacent The reciprocal trigonometric functions of the angle θ are formed by exchanging the terms in each ratio. cosecant - csc θ = hypotenuse/opposite secant - sec θ = hypotenuse/adjacent cotangent - cot θ = adjacent/opposite

What is tan (-5π/6)?

tan (-5π/6) = tan 210 Q3 --> tan is positive tan 210 - 180 = tan 30 tan 30 === √3/3

What is tan 675?

tan 675 = tan 315 Q4 360 - 315 = 45 - tan 45 === -1

Rewriting functions based off of the other functions

tan x = sin x/cos x cot x = cos x/sin x csc x = 1/sin x sec x = 1/cos x

QUICK CHART - NEED TO KNOW

x | 0 | 30 | 45 | 60 | 90 | -------------------------------------------------------------------- sin x | √3/2 = 0 | √1/2 = 1/2 | √2/2 | √3/2 | √4/2 = 1 | -------------------------------------------------------------------- cos x | √4/2 = 1 | √3/2 | √2/2 | √1/2 = 1/2 | √3/2 = 0 | -------------------------------------------------------------------- tan x | 0/1 = 0 | √3/3 | 1 | √3 | undefined | --------------------------------------------------------------------

Quadrantal Angles Table

| 0 | 90 | 180 | 270 | 360 | -------------------------------------------------------------------- sin θ | 0 | 1 | 0 | -1 | 0 | -------------------------------------------------------------------- cos θ | 1 | 0 | -1 | 0 | 1 | -------------------------------------------------------------------- tan θ | 0 | U | 0 | U | 0 | -------------------------------------------------------------------- tan = sine/cos