ACT Math : How to find a reference angle
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Example Questions
Example Question #1 : How To Find A Reference Angle
Which of the following is equivalent to cot(θ)sec(θ)sin(θ)
cot(θ)
–sec(θ)
0
tan(θ)
1
1
The first thing to do is to breakdown the meaning of each trig function, cot = cos/sin, sec = 1/cos, and sin = sin. Then put these back into the function and simplify if possible, so then (cos (Θ)/sin (Θ))*(1/cos (Θ))*(sin (Θ)) = (cos (Θ)*sin(Θ))/(sin (Θ)*cos(Θ)) = 1, since they all cancel out.
Example Question #1 : How To Find A Reference Angle
Using trigonometry identities, simplify sinθcos2θ – sinθ
None of these answers are correct
sin2θcosθ
cos3θ
cos2θsinθ
–sin3θ
–sin3θ
Factor the expression to get sinθ(cos2θ – 1).
The trig identity cos2θ + sin2θ = 1 can be reworked to becomes cos2θ – 1 = –sinθ resulting in the simplification of –sin3θ.
Example Question #1 : How To Find A Reference Angle
Using trig identities, simplify sinθ + cotθcosθ
secθ
tanθ
sin2θ
cos2θ
cscθ
cscθ
Cotθ can be written as cosθ/sinθ, which results in sinθ + cos2θ/sinθ.
Combining to get a single fraction results in (sin2θ + cos2θ)/sinθ.
Knowing that sin2θ + cos2θ = 1, we get 1/sinθ, which can be written as cscθ.
Example Question #2 : How To Find A Reference Angle
Simplify sec4Θ – tan4Θ.
tan2Θ – sin2Θ
secΘ + sinΘ
sinΘ + cosΘ
cosΘ – sinΘ
sec2Θ + tan2Θ
sec2Θ + tan2Θ
Factor using the difference of two squares: a2 – b2 = (a + b)(a – b)
The identity 1 + tan2Θ = sec2Θ which can be rewritten as 1 = sec2Θ – tan2Θ
So sec4Θ – tan4Θ = (sec2Θ + tan2Θ)(sec2Θ – tan2Θ) = (sec2Θ + tan2Θ)(1) = sec2Θ + tan2Θ
Example Question #2 : How To Find A Reference Angle
Evaluate the expression below.
At 
Cotangent is given by 
Now we can evaluate the expression.
Example Question #3 : Reference Angles
What is the reference angle of an angle that measures 3510 in standard position?
351
90
109
369
90
3600 – 3510 = 90
Example Question #2 : How To Find A Reference Angle
Simplify the following expression:
None of the answers are correct
tanΘ
cos2Θ
cscΘ
sin2Θ
sin2Θ
Convert cotΘ and secΘ to sinΘ and cosΘ and simplify the resulting complex fraction.
cotΘ = cosΘ secΘ = 1
sinΘ cosΘ
Example Question #3 : How To Find A Reference Angle
What is the reference angle for 
The reference angle is between 
To find the reference angle, we subtract 
Example Question #4 : How To Find A Reference Angle
In the unit circle above, if 
On the unit circle, (X,Y) = (cos Θ, sin Θ).
(cos Θ,sin Θ) = (cos 30º, sin 30º) = (√3 / 2 , 1 / 2)
Example Question #10 : How To Find A Reference Angle
What is the reference angle for 
A reference angle is the smallest possible angle between a given angle measurement and the x-axis.
In this case, our angle 
Thus, the reference angle for 
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