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Trigonometry: Fill in the Blank

Fill in the Blank 19 questions Mathematics & Statistics > Trigonometry by steven marone
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Fill in the Blank (19)

Question 1
On the unit circle, the x-coordinate of a point corresponding to an angle θ represents the cosine of that angle.
Missing Word
cosine
For any angle θ on the unit circle, the x-coordinate of the point where the terminal side intersects the circle is defined as cos(θ).
Question 2
The y-coordinate of a point on the unit circle corresponds to the sine of the angle formed with the positive x-axis.
Missing Word
sine
The sine of an angle θ is defined as the y-coordinate of the point where the terminal side of the angle intersects the unit circle.
Question 3
In the second quadrant, the sine function is positive while the cosine function is negative.
Missing Word
positive
In the second quadrant, y-coordinates are positive and x-coordinates are negative, meaning sine is positive and cosine is negative.
Question 4
The tangent of an angle is equivalent to the ratio of the sine of the angle to its cosine.
Missing Word
tangent
The quotient identity states that tan(θ) = sin(θ) / cos(θ).
Question 5
The fundamental trigonometric identity states that the square of the sine of an angle plus the square of its cosine equals one.
Missing Word
one
The Pythagorean identity is sin²(θ) + cos²(θ) = 1, derived from the equation of the unit circle x² + y² = 1.
Question 6
The cosecant function is the reciprocal of the sine function.
Missing Word
cosecant
Cosecant (csc θ) is defined as 1/sin θ.
Question 7
The period of both the sine and cosine functions is 2π radians.
Missing Word
period
The sine and cosine functions complete one full cycle over an interval of 2π radians.
Question 8
For the function y = A sin(Bx + C) + D, the absolute value of A represents the amplitude.
Missing Word
amplitude
The amplitude of a sinusoidal function is half the distance between its maximum and minimum values, which is given by |A|.
Question 9
To solve a triangle given two angles and a non-included side, the Law of Sines is the most appropriate tool.
Missing Word
Sines
The Law of Sines is used when you have Angle-Angle-Side (AAS) or Angle-Side-Angle (ASA) information.
Question 10
When all three sides of a triangle are known, the Law of Cosines can be used to find any of the angles.
Missing Word
Cosines
The Law of Cosines is used when you have Side-Side-Side (SSS) or Side-Angle-Side (SAS) information.
Question 11
The graph of the tangent function has vertical asymptotes where the cosine of the angle is zero.
Missing Word
asymptotes
Tan(θ) = sin(θ)/cos(θ), so it is undefined when cos(θ) = 0, leading to vertical asymptotes at those points.
Question 12
At an angle of 0 radians on the unit circle, the coordinates are (1, 0).
Missing Word
radians
0 radians corresponds to the point (1,0) on the unit circle, where cos(0)=1 and sin(0)=0.
Question 13
The sine of π/6 radians is 1/2.
Missing Word
1/2
On the unit circle, the y-coordinate at π/6 (30 degrees) is 1/2.
Question 14
The cosine of π/4 radians is √2/2.
Missing Word
√2/2
On the unit circle, the x-coordinate at π/4 (45 degrees) is √2/2.
Question 15
The tangent of π/4 radians is 1.
Missing Word
1
Tan(π/4) = sin(π/4) / cos(π/4) = (√2/2) / (√2/2) = 1.
Question 16
The vertical shift in a sinusoidal function is represented by the constant D in the equation y = A sin(Bx + C) + D.
Missing Word
vertical
The value of D shifts the entire graph up or down.
Question 17
The cosine function is an even function, meaning cos(-θ) = cos(θ).
Missing Word
even
An even function exhibits symmetry about the y-axis, and its value for a negative input is the same as for the positive input.
Question 18
The ambiguous case in the Law of Sines occurs when given two sides and a non-included angle (SSA), potentially leading to zero, one, or two possible triangles.
Missing Word
ambiguous
SSA is the only case where the Law of Sines might yield multiple valid triangles or no triangle at all.
Question 19
Unlike sine and cosine, the period of the tangent function is π radians.
Missing Word
π
The tangent function repeats its values every π radians, half the period of sine and cosine.

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