11.2 Trigonometric Functions

+ and - coterminal angles of 5π/4?

13π/4 and -3π/4

1 radian

180/π convert from degree to radians: degrees×π/180°

convert π/9 from radians to degrees...

20°

coterminal of 640°?

280°

coterminal of -435°?

285°

-7π/9 reference angle

2π/9

Range of csc^-1(x)

(-π/2, 0] ∪ [0, π/2)

sec range

(-∞, -1] U [1, ∞)

Domain of sec^-1(x)

(-∞, -1] ∪[1, ∞)

Domain of csc^-1(x)

(-∞,-1] ∪ [1, ∞)

Range of sec^-1(x)

(0, π/2] ∪ [π/2, π)

a function will have an inverse only if it is a one-to-one function and satisfies the horizontal line test, T or F?

True

coterminals, 17π/36, 161π/36?

Yes

Domain of sin^-1(x)

[-1,1]

cos range

[-1,1]

sin range

[-1,1]

Domain of cos^-1(x)

[-1,1], using the graph...

Range of sin^-1(x)

[-pi/2, pi/2]

Range of tan^-1(x)

[-pi/2, pi/2]

Domain of tan^-1(x)

[-∞,∞]

Range of cos^-1(x)

[0, pi], using the graph

Unit Circle

a circle with a radius of 1, centered at the origin

right triangle, c? a=8√3, b=16

c= 8

right triangle, c? a=√3, b=9

c=√3

standard position

- an angle is in standard postion if its vertex is located at the origin and one ray is on the positive x-axis. The ray on the x-axis is called the initial side, and the other ray is called the terminal side.

derivative of csc^-1(x), where x∈(-∞, -1) ∪ (1, ∞)?

-1/(absX×√(x²-1)

derivative of cos^-1(x), where x∈(-1,1)?

-1/(√1-x²)

y=sin^-1x <=> siny=x

-1≤siny≤1 and -1≤x≤1

tan (-a)

-tan a odd function

y=tan^-1x <=> tany=x for

-π/2<y<π/2

-cos(90)

0

distance from (1,0) to point (½,√3/2), units in km...?

1 km

proof of cos^-1(x) dx?

1. cos(cos^-1x)=x. cos^-1(cosx)=x 2. f(x)=cosx. g(x)=cos^-1(x) 3. g'(x)=1/f'(g(x)) = 1/-sin(cos^-1x) 4. y=cos^-1(x) => x=cos(y) - using #3, the denominator in the derivative becomes info in #4 5. Recall that cos²y+sin²y=1. =>siny=√(1-cos²y) 6. sin(cos^-1x)=√(1-sin²y)=√(1-x²) 7. d/dx(cos^-1x)= -1 √(1-x²)

proof of tan^-1(x) dx?

1. limₓ͢. ∞ tan^-1x =π/2. and limₓ͢.-∞ tan^-1x =-π/2 2. the tangent and inverse tangent functions and inverse functions so... tan(tan^-1x)=x. tan^-1(tanx)=x ∴d/dx of inv tan function f(x)=tanx. g(x)=tan^-1x 3. g'(x)=1/f'(g(x)) =1/sec²(tan^-1x) 4. y=tan^-1x => tany=x 5. the denominator is sec²(tan^-1x)=sec²y 6. cos²y+sin²y=1 7. divide by cos²y, 1+tan²y=sec²y 8. sec²(tan^-1x)=sec²y=1+tan²y sec²(tan^-1x)= 1+tan²y=1+x² 9. d/dx(tan^-1x)=1/1+x²

proof of sin^-1(x) dx?

1. sin(sin^-1x)=x. sin^-1(sinx)=x 2. f(x)=sinx. g(x)=sin^-1(x) 3. g'(x)=1/f'(g(x)) = 1/cos(sin^-1x) 4. y=sin^-1(x) => x=sin(y) - using #3, the denominator in the derivative becomes info in #4 5. Recall that cos²y+sin²y=1. => cosy=√(1-sin²y) 6. cos(sin^-1x)=√(1-sin²y)= √(1-x²)

derivative of sec^-1(x), where x∈(-∞, -1) ∪ (1, ∞)?

1/(absX×√(x²-1)

derivative of sin^-1(x), where x∈(-1,1)?

1/(√1-x²)

derivative of tan^-1(x), where x∈R?

1/x²+1

coterminal of -330°?

30°

reference angle for 510°

30°

Q4 reference angle

360- ϴ₄

differentiate...?f(t)=4cos^-1(t)-10tan^-1(t)

4(-1/√(1-x²)-10(1/1+x²)

arccos(455/600)=

40.68°

31π/9 reference angle

4π/9

230° reference angle

50°

coterminal of -19π/12?

5π/12

coterminal of 11π/3?

5π/3

find the radius of the circle given the Arc Length of 3 pi and central angle of Pi / 2

6 = 2/π*3π

reference angle for -250°

70°

-25π/18 reference angle

7π/18

-29π/18 reference angle

7π/18

coterminal of 15π/4?

7π/4

reference angle for 640°

80°

+ and - coterminal angles of 25π/36?

97π/36 and -47π/36

equilateral triangle

A triangle with three congruent sides

arc length

S

#83- A person walks along a circular track of radius 1.0 kilometer, centered at the origin of an X-y coordinate plane. If the person walks from point (1,0) to the point (½, √3/2) approximately how far does a person walk along the track?

Competency 11.2 this question requires the examinee to apply trigonometric functions to solve problems involving the unit circle. 1. Distance between two points: √(√3/2-0)²+(½-1)²= √¾+¼=√1=1 2. △OBC is an equilateral triangle 3. ∴arc⌒CB=Rϴ=1×π/3= 1.05km

Inverse Trig Functions

Cosecant, secant, cotangent

trigonometric functions

SOH CAH TOA (sine is opposite over hypotenuse, etc.)

Find the height of a tree (HT of a right triangle HAT) if the angle of elevation (<A) measures 40 degrees and the distance to the tree (HA) is 36 meters.

In a right triangle, the ratio of the length of the side opposite of a 40 degree angle divided by the length of the side adjacent to a 40 degree angle, called the tangent of 40 degrees (tan 40 degrees) is about .84. Therefore, HT/HA = .84 HT/36 = .84 HT = (36)(.84) HT = 30.24, the height of the tree is approximately 30 meters.

coterminals, 185°, -545°?

No

180- ϴ₂

Q2 reference angle

Reference angle - find the reference angle of 8π/3 in radians?

Solution: the given angle is greater than 2π. 1. We find the coterminal angle by subtracting 2π from it. 8π/3 - 2π = 2π/3 2. This angle does not lie between 0 and pi/2. Hence, it is not the reference angle of the given angle. 3. Finding the reference angle, is 2π/3 closest to π or 2π and by how much? Clearly 2π/3 is close to π by π - 2π/3 = π/3. Therefore, the reference angle of 8π/3 is π/3.

circumference

The distance around a circle, 2πr

Hypotenuse of right triangle

The side opposite the 90° < (or 2X) and longest side of the right angle in a right triangle.

isosceles triangle

a triangle with at least two congruent or equal sides

scalene triangle

a triangle with no congruent sides

right triangle, a? b=10, c=20

a=10√3

cos domain

all real numbers

cot range

all real numbers

sin domain

all real numbers

tan range

all real numbers

cot domain

all real numbers except (nπ...n is an integer) where sin is 0

sec domain

all real numbers except (π/2+kπ...k is an integer) where cos is 0

tan x domain

all real numbers except (π/2+kπ...k is an integer) where cos is 0

cos(-a)

cos a even function

cot (-a)

cot a odd function

F(-x) = (-x)^6-(-x)^2+7, even, odd or neither...?

even, Since x^6+x+7 is an even function, the function is even.

y=arcsinx at x= -1

f(-1)=arcsin(-1) f(-1)=-π/2

odd function

f(-x)=-f(x) symmetric about origin

even function

f(-x)=f(x)

every math function has an inverse or opposite, T or F

false. It is easy to see the function f(x) is going to have an inverse, the f(x) takes on the same false value twice. It is not a one-to-one function.

y=cos^-1x <=> cosy=x

for 0≤y≤π -1≤x≤1 because -1≤cosy≤1

central angle

formed at the center of a circle due to the intersection of any 2 radii within a circle....

inverse function formula

g'(x)=1/[f'(g(x)]

Radian measure formula

s=r(theta) 1 radian = 180/π degrees, 1 degree = π/180° radians

shortest leg of a right triangle

side opposite 30° or X or divide by √3

2nd longest leg of a right triangle

side opposite of 60° < , (X = shorter leg × √3)

finding the reference angle

the acute angle formed by the terminal side of an angle in standard position and the x-axis I- reference angle= Angle II- reference angle = 180-angle III- reference angle= angle - 180 IV- reference angle= 360-angle

reference angle

the angle made with the terminal arm of the standard angle and the x-axis

coterminal angle

two angles in standard position that have the same terminal side If ϴ is any angle, then ϴ+ n (360) is coterminal with ϴ for all non zero integer n. For positive < ϴ, the coterminal < can be found by: ϴ+360°

inverse cosine function

x=cosy

Inverse Cosecant Function

x=cscy

Inverse Secant Function

x=secy

inverse sine function

x=sin(y)

inverse tangent function

x=tany

evaluate cos^-1(-√2/2)

y = 3π/4

differentiate √z×sin^-1(z)

½z^-½sin^-1(z)+√z/(√1+z²)

reference angle for -13π/12

π/12

coterminal of -35π/18?

π/18

reference angle for -19π/18

π/18

evaluate tan^-1

π/4

siny = ½

π/6

Quarter 1 reference angle

ϴ = ϴ₁ in Q1

Q3 reference angle

ϴ₃-180°