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Common Core Standard HSF-TF.A.4 Questions

(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

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Grade 11 Trigonometry CCSS: HSF-TF.A.4

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What is the value of sin(-θ)?
  1. -x1y1
  2. x1y1
  3. -y1x1
  4. -y1
Grade 11 Trigonometry CCSS: HSF-TF.A.4

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The period of a periodic function is the smallest value of T such that f(x+T)=f(x) for all values of x in the domain of f. Using the information in the previous questions, which of the following gives the best reasoning as to why the period of the sine function is 2π?
  1. Since values of T less than 2π weren't valid, 2π must work.
  2. Using T=2π and trying a few values of θ, such as 3π7 and 8π11, the equation sin(θ+2π)=sin(θ) is valid. Therefore, it is true for all values of θ.
  3. For T=2π, θ+T and θ are coterminal angles for any value of θ. Therefore, referring to the unit circle, sin(θ)=b and sin(θ+2π)=b, and therefore sin(θ+2π)=sin(θ).
  4. There are no other values of T such that sin(x+T)=sin(x).
Grade 11 Trigonometry CCSS: HSF-TF.A.4

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What conclusion from the previous questions can be reached?
  1. sin(θ)=y1 and sin(-θ)=-x1 for 0θπ2.
  2. sin(θ)=y1x1=1sin(-θ) for 0θπ2.
  3. sin(θ)=y1x1=-sin(-θ) for 0θπ2.
  4. sin(θ)=y1=-sin(-θ) for 0θπ2.
Grade 11 Trigonometry CCSS: HSF-TF.A.4

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The period of the cosine function is also 2π, and showing this is true is similar to the process used for the sine function. However, the period of the tangent function is π. Which of the following equations correctly shows why the period of tangent is π?
  1. tan(θ+π)=sin(θ+π)cos(θ+π)=-sin(θ)-cos(θ)=sin(θ)cos(θ)=tan(θ)
  2. tan(θ+π)=sin(θ+π)cos(θ+π)=sin(θ)+πcos(θ)+π=sin(θ)cos(θ)=tan(θ)
  3. tan(θ+π)=sin(θ+π)cos(θ+π)=sin(θ)+sin(π)cos(θ)+cos(π)=sin(θ)+0cos(θ)+0=tan(θ)
  4. tan(θ+π)=tan(θ)+tan(π)=tan(θ)
Grade 11 Trigonometry CCSS: HSF-TF.A.4

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Let π2<απ be an angle in standard position, whose terminal arm intersects circle A at (-x2,y2). Can the same conclusion be reached concerning sin(α) as was reached for sin(θ), with the same reasoning? Why?
  1. Yes, since neither conclusion depends upon the x-values (it doesn't matter if they're positive or negative).
  2. Yes, since sin(α)=sin(θ).
  3. No, because the reasoning applied is only valid for angles in the first quadrant.
  4. No, because the sine function is only positive in the first quadrant.
Grade 11 Trigonometry CCSS: HSF-TF.A.4

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Looking at π<β3π2 and 3π2<γ2π, where both angles are in standard position, and their terminal arms intersect circle A at (-x3,-y3) and (x4,-y4) respectively, which of the following explains why the sine of an angle equals the negative sine of the negative of that angle for angles between π and 2π?
  1. Since sine is periodic, whatever is true for the sine function for values between 0 and π must also be true for values between π and 2π.
  2. Using the same reasoning as before, for angle β it follows that sin(β)=-y3=-sin(-β) and for angle γ that sin(γ)=-y4=-sin(-γ).
  3. As seen before, the sine of an angle in one quadrant is always the opposite of the sine of an angle in an adjacent quadrant. Thus sin(β)=-sin(-β) and sin(γ)=-sin(-γ).
  4. Looking at angles between π and 2π, the sine of these angles is always negative. Therefore, sin(β)=-sin(-β) and sin(γ)=-sin(-γ).
Grade 11 Trigonometry CCSS: HSF-TF.A.4

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Which of the following reasons allows one to conclude that sin(Θ) is odd?
  1. It has been shown that sin(-ω)=-sin(ω) for 0ω2π. Since sine is periodic with a period of 2π, this result must be true for ω.
  2. It has been shown that sin(-ω)=-sin(ω) for 0ω2π. Since -1sin(ω)1 is true not only for values between 0 and 2π, but for all real values of ω, it follows that sin(-ω)=-sin(ω) for ω.
  3. Since the values of sine from the first quadrant are simply repeated in the other quadrants (except for being sometimes negative), all that one needed to confirm was that sin(-ω)=-sin(ω) for 0ωπ2, and then it must be true for ω.
  4. Using circle A, it has been seen that, for 0ω2π, the sine of ω is always greater than or equal to zero, while the sine of -ω is always less than or equal to zero. Therefore, since sine is periodic with period 2π, this result must be true for ω.
Grade 11 Trigonometry CCSS: HSF-TF.A.4

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Which of the following statements best explains the symmetry of cos(Θ)? Assume that Θ can be interpreted as an angle in standard position, whose terminal arm intersects circle A at the point (X1,Y1), where X1,Y1 are real numbers.
  1. Since we know that sine and cosine are the same function, except that cosine is different by a phase shift of π2, therefore the same properties must apply to both functions. Thus, cosine is an odd function.
  2. Using the same reasoning as was applied to the sine function in the previous questions, it can be seen that that cos(Θ)=X1 and cos(-Θ)=-X1. Therefore cos(Θ)=-cos(-Θ), showing that cosine is odd.
  3. For any given angle Θ, the value of cos(Θ) can be interpreted as X1. Using similar reasoning as was used in previous questions, it can be seen that cos(-Θ)=X1, where X1 will always be of the same sign. Hence, cos(Θ)=cos(-Θ), showing that the cosine function is even.
  4. Looking at the unit circle, the sine of any angle Θ will be Y1 and the cosine of the angle will be X1. The properties of the sine function can therefore be seen in the properties of the cosine function, if we simply reflect the coordinate axis about the line y=x. Therefore, an odd function will become an even function, and thus cosine is an even function.
Grade 11 Trigonometry CCSS: HSF-TF.A.4

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Using the same reasoning as was done with the sine function, it can be shown that the tangent function is odd. However, there are values for which the tangent function is undefined (its domain is not all real numbers). How is this possible, given that the unit circle has no undefined values on its circumference?
  1. Since the tangent function has a period of π instead of 2π, it is actually associated with the unit semicircle centered at the origin. Therefore, because of the discontinuity of it being a semicircle instead of a full circle, the undefined values arise.
  2. The values of the tangent function do not occur on circle A. Referring to a right triangle, the tangent of either acute angle is the opposite side divided by the adjacent side. Since this value does not include the hypotenuse, the points on circle A are actually only an approximation of the tangent function values. Therefore, although the tangent function has undefined values, these do not arise in the approximated values of circle A.
  3. Since the values of tan(Θ) can be represented as Y1X1, the values where X1=0 are undefined and correspond to the domain restrictions of the tangent function.
  4. Since the values of tan(Θ) can be represented as X1Y1, the values where Y1=0 are undefined and correspond to the domain restrictions of the tangent function.
Grade 11 Trigonometry CCSS: HSF-TF.A.4

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For this investigation of trigonometric functions and their symmetry, the unit circle was used (circle A). Why was the unit circle used, and not a circle with a different radius (not equal to one unit)?
  1. Because the absolute value of the maximum and minimum values of the sine and cosine functions are 1.
  2. The circumference of a circle is C=2πr. If the radius is not equal to 1, the circumference would be greater than or less than 2π. Since the period of sine and cosine is 2π, this would mean that the values of these functions on the circle would not fit.
  3. It simplifies the math, allowing the coordinates of the points on the unit circle to be equal to the sine and cosine of angles in standard position.
  4. There is no particular reason, it is simply convention to use the unit circle. All the math and reasoning would have been exactly the same using a circle centered at the origin with a different radius.
Grade 11 Trigonometry CCSS: HSF-TF.A.4
Which of the following are equal to sin(π4)? Choose all that apply.
  1. sin(-7π4)
  2. sin(17π4)
  3. sin(13π4)
  4. sin(5π4)
Grade 11 Trigonometry CCSS: HSF-TF.A.4
Which of the following are equal to cos(π12)? Choose all that apply.
  1. cos(-35π12)
  2. cos(-π12)
  3. cos(25π12)
  4. cos(97π12)
Grade 11 Trigonometry CCSS: HSF-TF.A.4
Which of the following are equal to tan(5π7)? Choose all that apply.
  1. tan(2π7)
  2. tan(-2π7)
  3. tan(12π7)
  4. tan(18π7)
Grade 11 Trigonometry CCSS: HSF-TF.A.4
Which of the following are equal to sin(7π3)? Choose all that apply.
  1. sin(4π3)
  2. sin(π3)
  3. sin(-5π3)
  4. sin(-17π3)
Grade 11 Trigonometry CCSS: HSF-TF.A.4
Which of the following are equal to cos(4π9)? Choose all that apply.
  1. cos(-5π9)
  2. cos(13π9)
  3. cos(22π9)
  4. cos(-23π9)
Grade 11 Trigonometry CCSS: HSF-TF.A.4
Which of the following are equal to tan(13π3)? Choose all that apply.
  1. tan(10π3)
  2. tan(2π3)
  3. tan(-π3)
  4. tan(-11π3)
Grade 11 Trigonometry CCSS: HSF-TF.A.4

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For -θ, where would the terminal arm intersect circle A?
  1. (-x1,-y1)
  2. (-x1,y1)
  3. (x1,-y1)
  4. (x1,y1)
Grade 11 Trigonometry CCSS: HSF-TF.A.4

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