Trig final

The value of tan^-1X is an angle in the interval

(-pi/2,pi/2)

What is the range of y=tan^−1x​?

(-pi/2,pi/2)

Sec T=

1/x

Csc T=

1/y

Isosceles

A triangle that has 2 equal sides.

Equilateral

A triangle with three congruent sides

Area of a sector formula

A= 1/2 * r^2 * theta Ex: 1/2 * 90 feet^2 * 1/3= 1350 ft^2

Finding central angle

A=1/2 * r^2 * theta EX. A= 70ft; R= 10ft 2 (70)= 2 [1/2(10^2) theta] 140= 100 theta theta= 7/5 radians

Pythagorean Theorem

A^2+B^2= C^2 OR X^2+Y^2= R^2

If the terminal side of an angle lies in Quadrant I​, then

All of the trigonometric functions are positive.

Which of the following statements is not true concerning radian​ measure?

An angle in standard position having a radian measure of theta = -11pi/6 has a terminal side that lies in Quadrant IV.

Which of the following statements best describes an angle that is in standard​ position?

An angle is in standard position if the vertex is at the origin of a rectangular coordinate system and the initial side lies along the positive​ x-axis.

For any real number​ t, if​ P(x,y) is a point on the unit circle corresponding to​ t, then which of the following does not accurately define a trigonometric​ function?

Cot t= y/x, x =/ 0

Properties of graph y=cotx

D: R: (−∞,∞)​ Period: pi Odd y intercept: None Zeros of the form ((2N+1)/2)pi, where n is an integer Every halfway point has a y-coordinate of: −1 or 1​ Principle cycle: (0,pi)

Properties of graph y=cosx

D: (-infinity,infinity) All real numbers R: [-1,1] Period: 2pi Even y intercept: 1 X intercept: (2n+1)*pi/2, where n is an integer relative max: X=2piN, where N is an integer relative min: X=Pi+ 2piN, where N is an integer Zeros of the form ((2N+1)/2)pi, where n is an integer

Properties of graph y=sinx

D: (-infinity,infinity) All real numbers R: [-1,1] Period: 2pi Odd y intercept: 0 X intercept: npi, where n is an integer relative max: x= pi/2 + 2piN, where N is an integer relative min: 3pi/2 + 2piN, where N is an integer Zeros of the form: Nπ​, where n is any​ integer

Properties of graph y=tanx

D: (-infinity,infinity) All real numbers, except odd integer multiples of pi/2 R: (−∞,∞)​ Period: pi Odd y intercept: 0 X intercept: Zeros of the form Nπ​, where n is any​ integer Every halfway point has a y-coordinate of: −1 or 1​ Principle cycle: (-pi/2,pi/2)

Properties of graph y=secx

D: all real numbers except odd integer multiples of π/2​ R: (−∞,−1]∪[1,∞) Period: 2pi Even relative max: x=πn where n is an odd integer Vertical asymptotes of the form: x= nπ/2 where n is an odd integer

Properties of graph y=cscx

D: all real numbers except odd integer multiples of π/2​,pi, R: (−∞,−1]∪[1,∞) Period: 2pi Odd relative max: x= -pi/2 + 2piN, where N is an integer Vertical asymptotes of the form: x=nπ where n is an integer

Which of the following statements best describes the term general​ angles?

General angles are angles that are not restricted in size and can be either positive​ angles, negative​ angles, or zero.

Which of the following statements is not true about the function y=Acos(Bx)​?

If B>0​, then the function y=Acos(−Bx) is equivalent to the function y=−Acos(Bx).

Which of the following statements is not ​true? Choose the correct answer below

If the central angle of a sector of a circle is θ=5° and the radius of the circle is r=2 ​cm, then the arc length of the sector of the circle is 10 cm. NOT 5 DEGREES BUT 5 X 180/PI

Which of the following statements is​ true?

If θ is an angle belonging to π/6, π/3, π/4 ​families, then the reference angle will be π/6, π/3, π/4​, respectively.

Graph of y=cos x

Quarter points: (0,1), (pi/2,0), (pi, -1), (3pi/2,0), (2pi, 1)

A 10 inch​ (in diameter) pizza is cut into various sizes. What is the area of a piece that was cut with a central angle of 32°​?

REMEMBER TO DIVIDE THE DIAMETER INTO 2 TO GET THE RADIUS

Length of arc

S= R*Theta

Which of the following statements describes the definition of amplitude of a sine or cosine​ function?

The amplitude is the measure of half the distance between the maximum and minimum values.

Which of the following statements is not true concerning angle​ measure?

The angle in standard position formed by rotating the terminal side of angle one complete counterclockwise rotation has a radian measure of π radians.

If theta= sine^-1x​, then which of the following statements best describes angle θ​?

The angle θ is an angle satisfying the inequality -π/ 2 ≤ θ ≤ π/2 having a terminal side lying in Quadrant​ I, Quadrant​ IV, on the positive​ x-axis, on the positive​ y-axis, or on the negative​ y-axis.

If θ=cos^−1x​, then which of the following statements best describes angle θ​?

The angle θ is an angle satisfying the inequality 0≤θ≤π having a terminal side lying in Quadrant​ I, Quadrant​ II, on the positive​ x-axis, on the positive​ y-axis, or on the negative​ x-axis

If θ=tan^−1x​, then which of the following statements best describes angle θ​?

The angle θ is an angle satisfying the inequality −π/2 < θ < π/2 having a terminal side lying in Quadrant​ I, Quadrant​ IV, or on the positive​ x-axis.

Which of the following statements is not ​true? Choose the correct answer below.

The area of a sector of a circle is given by the equation A= 1/2 * θ * r^2​, where θ is an angle given in degrees. NOT DEGREES, RADIANS

Which of the following statements is​ true?

The length of the leg opposite either of the π/4 angles of a special π/4​, π/4​, π/2 right triangle with a hypotenuse of SquareRoot 2 is equal to 1.

Which of the following statements is​ true about a special right triangle?

The length of the leg opposite the π/3 angle of a special π/6​, π/3, π/2 right triangle is equal to the square root of 3 times the length of the leg opposite the π/6 angle

If​ A, B, and C are constants such that B>1​, then which of the following statements is true about the graph of y=Acos(Bx−C)​?

The period is 2π/B and the phase shift is C/B.

Which of the following is not a characteristic of the sine​ function?

The sine function obtains a relative maximum at x equals x= π/2+πn where n is an integer.

Which is true about the unit circle?

The unit circle is symmetric about the​ x-axis, y-axis, and the origin.

Which of the following statements is not​ true?

There are infinitely many points that lie on the graph of the unit circle that have integer coordinates.

Which of the following angles does not belong to the π/4 family of​ angles?

Theta= 10pi/4

Which of the following statements is​ true?

The​ x-intercepts of y=tanx are the same as the​ x-coordinates of the center points of y=tanx.

Which of the following is not a characteristic of the cosine​ function?

The​ y-intercept is 0.

Which of the following statements is true concerning the conversion between degree and radian​ measure?

To convert from radians to​ degrees, multiply by 180 degrees and divide by π.

Which of the following statements best describes two coterminal​ angles?

Two angles in standard position are coterminal if they have the same terminal side

The graph y=sin(x+C) can be obtained by horizontally shifting each quarter point of y=sin(x) to the left C units.

Which of the following statement best describes the graph of y=sin(x+C) where Upper C greater than 0C>0​?

The value of sine^-1X is an angle in the interval

[-pi/2,pi/2]

The value of cos^-1X is an angle in the interval

[0,pi]

What is the domain of the restricted cosine function whose inverse function is y=cos^−1x​?

[0,pi]

Scalene

a triangle with no congruent sides

Cofunction identities (Cos)

cos (pi/2- theta) = sin theta

Which of the following is not a fundamental​ identity?

cos theta= tan theta/ sin theta

Cofunction identities (Cot)

cot (pi/2 - theta) = tan theta

Trigonometric expression (Cot)

cot theta= cos theta/ sin theta

Suppose that a right triangle has an acute angle θ and side lengths of​ hyp, opp, and adj. Which of the following does not accurately define a trigonometric​ function?

cot theta= opp/adj

Trigonometric expression (Cot and csc)

cot^2theta+1=csc^2theta

Cofunction identities (Csc)

csc (pi/2 - theta) = sec theta

Which of the following is not a valid​ equation?

csc pi/6 = cos pi/3

Trigonometric expression (Csc)

csc theta= 1/sin theta

Graph of y=sin x

quarter points: (0,0),(pi/2,1),(pi,0),(3pi/2,-1),(2pi,0)

Secant/Sec

r/x= Hyp/Adj

Cosecant/Csc

r/y= Hyp/Opp

Cofunction identities (Sec)

sec (pi/2 - theta) = csc theta

Trigonometric expression (Sec)

sec theta= 1/cos theta

Which of the following is not a cofunction​ identity?

sec theta= cos (pi/2 - theta)

Which of the following is NOT a fundamental​ identity?

sec^2theta +1= tan^2theta

Cofunction identities (Sin)

sin (pi/2-theta) = cos theta

Trigonometric expression (Sin and cos)

sin^2theta+cos^2theta=1

Cofunction identities (Tan)

tan (pi/2 - theta) = cot theta

If the terminal side of an angle θ lies in Quadrant​ III, then which of the following is​ true?

tan theta > 0, sec theta < 0

Trigonometric expression (Tan)

tan theta= sin theta/cos theta

Trigonometric expression (Tan and sec)

tan^2theta+1=sec^2theta

tan: the angle lies in the interval 0<theta<pi/2

terminal side: Quad 1

sin: the angle lies in the interval 0<theta<pi/2 (+)

terminal side: Quad 1 or positive y-axis

tan: the angle lies in the interval -pi/2<theta<0

terminal side: Quad 4

sin: the angle lies in the interval (-pi/2<theta<0) (-)

terminal side: Quad 4 or negative y-axis

cos: the angle lies in the interval pi/2<theta<pi

terminal side: Quadrant 2 or negative​ x-axis

cos: the angle lies in the interval 0<theta<pi/2

terminal side: Quadrant I or positive​ x-axis

If the terminal side of an angle lies in Quadrant IV​, then

the Cosine ​(and secant) functions are positive.

If the terminal side of an angle lies in Quadrant II​, then

the Sine ​(and cosecant) functions are positive.

If the terminal side of an angle lies in Quadrant III​, then

the Tangent ​(and cotangent) functions are positive.

Value of B

to graph divide period into 4 and then add that number to each x value.

Value of A

to graph multiple Y value of quarter points by A.

Cos T=

x

Cos t=

x/1

Cosine/Cos

x/r= Adj/Hyp

Cot T=

x/y

Cotangent/Cot

x/y= Adj/Opp

The standard form of the equation of the unit circle is

x^2+y^2=1

Sin T=

y

Sin t=

y/1

Sine/Sin

y/r= Opp/Hyp

Tan T=

y/x

Tangent/Tan

y/x= Opp/Adj

y=-2cos(-4x) is the same as

y=-2cos(4x)

y=sin(-pix) is the same as

y=-sin(pix)

Phase shift:

y=Asin(Bx+C) C/B

Amplitude

y=AsinX IAI absolute value of A so always positive. Changes range

Period

y=sin(BX) Period= 2pi/B as long as B is positive

Finding the principle cycle of y=Atan(Bx−C)+D​,

−π/2<Bx−C<π/2