Reference Angles - ACT Math
Card 0 of 12
What is the reference angle of an angle that measures 3510 in standard position?
What is the reference angle of an angle that measures 3510 in standard position?
3600 – 3510 = 90
3600 – 3510 = 90
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Which of the following is equivalent to cot(θ)sec(θ)sin(θ)
Which of the following is equivalent to cot(θ)sec(θ)sin(θ)
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.
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.
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In the unit circle above, if 
, what are the coordinates of 
?
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)
On the unit circle, (X,Y) = (cos Θ, sin Θ).
(cos Θ,sin Θ) = (cos 30º, sin 30º) = (√3 / 2 , 1 / 2)
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Simplify the following expression:
Simplify the following expression:
Convert cotΘ and secΘ to sinΘ and cosΘ and simplify the resulting complex fraction.
cotΘ = cosΘ secΘ = 1
sinΘ cosΘ
Convert cotΘ and secΘ to sinΘ and cosΘ and simplify the resulting complex fraction.
cotΘ = cosΘ secΘ = 1
sinΘ cosΘ
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Using trigonometry identities, simplify sinθcos2θ – sinθ
Using trigonometry identities, simplify sinθcos2θ – sinθ
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.
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.
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Using trig identities, simplify sinθ + cotθcosθ
Using trig identities, simplify sinθ + cotθcosθ
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.
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.
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Simplify sec4_Θ_ – tan4_Θ_.
Simplify sec4_Θ_ – tan4_Θ_.
Factor using the difference of two squares: _a_2 – _b_2 = (a + b)(a – b)
The identity 1 + tan2_Θ_ = sec2_Θ_ which can be rewritten as 1 = sec2_Θ_ – tan2_Θ_
So sec4_Θ_ – tan4_Θ_ = (sec2_Θ_ + tan2_Θ_)(sec2_Θ_ – tan2_Θ_) = (sec2_Θ_ + tan2_Θ_)(1) = sec2_Θ_ + tan2_Θ_
Factor using the difference of two squares: _a_2 – _b_2 = (a + b)(a – b)
The identity 1 + tan2_Θ_ = sec2_Θ_ which can be rewritten as 1 = sec2_Θ_ – tan2_Θ_
So sec4_Θ_ – tan4_Θ_ = (sec2_Θ_ + tan2_Θ_)(sec2_Θ_ – tan2_Θ_) = (sec2_Θ_ + tan2_Θ_)(1) = sec2_Θ_ + tan2_Θ_
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Evaluate the expression below.
Evaluate the expression below.
At 
, sine and cosine have the same value.
Cotangent is given by 
.
Now we can evaluate the expression.
At
Cotangent is given by
Now we can evaluate the expression.
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What is the reference angle for 
?
What is the reference angle for
The reference angle is between 
and 
, starting on the positive 
-axis and rotating in a counter-clockwise manor.
To find the reference angle, we subtract 
for each complete revolution until we get a positive number less than 
.
is equal to two complete revolutions, plus a 
angle. Since 
is in Quadrant II, we subtract it from 
to get our reference angle:
The reference angle is between
To find the reference angle, we subtract
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What is the reference angle for 
?
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 
lies in Quadrant I, so the angle is its own reference angle.
Thus, the reference angle for 
is 
.
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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What is the reference angle for 
?
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 
lies in Quadrant III, so the angle is found by the formula 
.
Thus, the reference angle for 
is 
.
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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What is the reference angle for 
?
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 
lies in Quadrant II, so we can find our reference angle using the formula
.
Thus, the reference angle for 
is 
.
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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