Using the periodicity property of cosine, , what is ?
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Precalculus Quiz
Practice Symmetry And Periodicity Of Trigonometric Functions in Precalculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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Using the periodicity property of cosine, cos(θ+2π)=cos(θ), what is cos(613π)?
This quiz focuses on Symmetry And Periodicity Of Trigonometric Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for Precalculus.
Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.
Using the periodicity property of cosine, cos(θ+2π)=cos(θ), what is cos(613π)?
Explanation: This question tests understanding of the periodicity of trigonometric functions. The cosine function is periodic with period 2π, meaning cos(θ + 2π) = cos(θ) for all θ. Since 13π/6 = π/6 + 2π, we can write cos(13π/6) = cos(π/6 + 2π) = cos(π/6) = √3/2. Choice A is correct because it uses the period 2π correctly to simplify 13π/6 to π/6, then evaluates cos(π/6) = √3/2. Choice B incorrectly gives the negative value -√3/2, perhaps confusing this angle with one in a different quadrant. To evaluate trig functions at angles beyond 2π, first use periodicity to reduce to the range [0, 2π), then evaluate using special angles or the unit circle.
Using the periodicity property of sine, sin(θ+2π)=sin(θ) for all θ. What is sin(49π)?
Explanation: This question tests understanding of the periodicity of trigonometric functions. The sine function is periodic with period 2π, meaning sin(θ + 2π) = sin(θ) for all θ. Since 9π/4 = π/4 + 2π, we can write sin(9π/4) = sin(π/4 + 2π) = sin(π/4) = √2/2. Choice B is correct because it uses the period correctly to simplify 9π/4 to π/4, then evaluates sin(π/4) = √2/2. Choice A incorrectly gives the negative value, perhaps confusing this with a different quadrant or applying an odd property where none is needed. To evaluate trig functions at angles beyond 2π, first use periodicity to reduce to the range [0, 2π), then use reference angles if needed.
If f(x)=sin(x)+cos(x) and g(x)=sin(x)−cos(x), which statement about the symmetry properties of these functions is correct?
Explanation: To determine symmetry, we test if f(−x)=f(x) (even), f(−x)=−f(x) (odd), or neither. For f(x)=sin(x)+cos(x): f(−x)=sin(−x)+cos(−x)=−sin(x)+cos(x). This equals neither f(x) nor −f(x). For g(x)=sin(x)−cos(x): g(−x)=sin(−x)−cos(−x)=−sin(x)−cos(x). This equals neither g(x) nor −g(x). Therefore, both functions are neither even nor odd.
A function g(x) satisfies g(x+4π)=g(x) for all x, and g(x)=sin(kx) for some constant k>0. What is the smallest possible positive value of k?
Explanation: For g(x)=sin(kx) to have period 4π, we need k2π=4π. Solving: 2π=4πk, so k=21. We can verify: g(x+4π)=sin(21(x+4π))=sin(2x+2π)=sin(2x)=g(x). The other values of k would give different periods: k=41 gives period 8π, k=1 gives period 2π, and k=2 gives period π.
Using the periodicity property of cosine on the unit circle (one full rotation), what is cos(θ+2π)?
Explanation: This question tests understanding of the periodicity of trigonometric functions. The sine and cosine functions are periodic with period 2π, meaning sin(θ + 2π) = sin(θ) and cos(θ + 2π) = cos(θ), while tangent has a shorter period of π, meaning tan(θ + π) = tan(θ). Since cosine has period 2π, we can write θ + 2π = θ + (one period), so cos(θ + 2π) = cos(θ). Choice A is correct because it applies the periodicity property correctly, recognizing that adding 2π (one full rotation on the unit circle) returns us to the same cosine value. Choice C incorrectly suggests adding π instead of recognizing that the function value repeats after 2π, which would give cos(θ + π) = -cos(θ), not cos(θ). To evaluate trig functions at negative angles or angles beyond 2π, first use periodicity to reduce to the range [0, 2π), then use even/odd properties if the angle is negative, and finally use reference angles if needed.
If sin(6π)=21, then based on sine being an odd function, what is sin(−6π)?
Explanation: This question tests understanding of the even/odd symmetry properties of trigonometric functions. Sine and tangent are odd functions, meaning sin(-θ) = -sin(θ) and tan(-θ) = -tan(θ), while cosine is an even function, meaning cos(-θ) = cos(θ). For angle π/6, we use the odd property of sine: sin(-π/6) = -sin(π/6) = -1/2. Choice B is correct because it applies the odd property correctly to the given value. Choice A incorrectly treats sine as an even function, using sin(-θ) = sin(θ), when sine is actually odd with sin(-θ) = -sin(θ). Key to symmetry and periodicity: remember that cosine is EVEN (cos(-x) = cos(x)), while sine and tangent are ODD (sin(-x) = -sin(x), tan(-x) = -tan(x)), and that sine and cosine repeat every 2π while tangent repeats every π. To evaluate trig functions at negative angles or angles beyond 2π, first use periodicity to reduce to the range [0, 2π), then use even/odd properties if the angle is negative, and finally use reference angles if needed.
The function h(x)=∣sin(x)∣ has which of the following properties?
Explanation: When analyzing transformed trigonometric functions, you need to examine how absolute value affects both the period and symmetry of the original function. To find the period of h(x)=∣sin(x)∣, consider what the absolute value does: it reflects all negative portions of sin(x) above the x-axis. Since sin(x) is negative on intervals like (π,2π), these portions get flipped upward. This creates a repeating pattern every π units instead of 2π. You can verify this: h(x+π)=∣sin(x+π)∣=∣−sin(x)∣=∣sin(x)∣=h(x). So the period is π. For symmetry, check if h(−x)=h(x) (even) or h(−x)=−h(x) (odd). Since h(−x)=∣sin(−x)∣=∣−sin(x)∣=∣sin(x)∣=h(x), the function has even symmetry about the y-axis. Looking at the wrong answers: Choice B incorrectly states the period as 2π - this would be true for the original sin(x), but absolute value creates more frequent repetition. Choice C suggests odd symmetry, but ∣sin(x)∣ is always non-negative, so h(−x)=−h(x) is impossible since the right side would be non-positive. Choice D correctly identifies the period as π but wrongly claims the function has neither even nor odd symmetry. Study tip: When you see absolute value applied to periodic functions, expect the period to be halved and the function to become even (symmetric about the y-axis) since absolute value eliminates negative outputs.
Using the periodicity property, sin(θ+2π)=sin(θ), what is sin(617π)?
Explanation: This question tests understanding of the periodicity of trigonometric functions. The sine and cosine functions are periodic with period 2π, meaning sin(θ + 2π) = sin(θ) and cos(θ + 2π) = cos(θ), while tangent has a shorter period of π, meaning tan(θ + π) = tan(θ). Since sine has period 2π, we can write 17π/6 = 5π/6 + 2π, so sin(17π/6) = sin(5π/6) = 1/2. Choice B is correct because it uses the period correctly. Choice A incorrectly uses the wrong period, claiming the period of sine is π when it's actually 2π. To evaluate trig functions at angles beyond 2π, first use periodicity to reduce to the range [0, 2π), then use reference angles if needed. When working with properties, apply them systematically: first reduce using periodicity (add/subtract multiples of the period), then apply even/odd symmetry if negative, then evaluate using special angles or the unit circle.
Based on the unit circle, the point at angle θ is (cos(θ),sin(θ)) and the point at angle −θ reflects across the x-axis. If sin(6π)=21, what is sin(−6π)?
Explanation: This question tests understanding of the even/odd symmetry properties of trigonometric functions. Sine and tangent are odd functions, meaning sin(−θ)=−sin(θ) and tan(−θ)=−tan(θ), while cosine is an even function, meaning cos(−θ)=cos(θ). On the unit circle, the point at angle θ has coordinates (cos(θ),sin(θ)), and the point at angle −θ has coordinates (cos(−θ),sin(−θ))=(cos(θ),−sin(θ)), reflecting across the x-axis. For angle 6π, we use the odd property of sine: sin(−6π)=−sin(6π)=−(21)=−21. Choice B is correct because it applies the odd property correctly with the given value. Choice A incorrectly treats sine as an even function, using sin(−θ)=sin(θ), when sine is actually odd with sin(−θ)=−sin(θ). The unit circle provides geometric intuition: reflecting across the x-axis (going from θ to −θ) keeps the x-coordinate (cosine) the same but flips the y-coordinate (sine), explaining why cosine is even and sine is odd.
Using periodic behavior, the smallest positive number p such that tan(θ+p)=tan(θ) for all θ is called the period of tangent. What is the period of tan(θ)?
Explanation: This question tests understanding of the periodicity of trigonometric functions. The sine and cosine functions are periodic with period 2π, meaning sin(θ + 2π) = sin(θ) and cos(θ + 2π) = cos(θ), while tangent has a shorter period of π, meaning tan(θ + π) = tan(θ). The period of tangent is π because this is the smallest positive value p such that tan(θ + p) = tan(θ) for all θ; tangent's vertical asymptotes repeat every π, completing one full cycle in that interval. Choice B is correct because it uses the period correctly, identifying the fundamental repeat distance for tangent. Choice A uses the wrong period, claiming the period of tangent is 2π when it's actually π. For tangent, remember its period is π (not 2π) because tan(θ) = sin(θ)/cos(θ), and both sine and cosine change sign when moving π radians, making their ratio the same. When working with properties, apply them systematically: first reduce using periodicity (add/subtract multiples of the period), then apply even/odd symmetry, then evaluate using special angles or the unit circle.
On the unit circle, the point at angle θ is (cos(θ),sin(θ)) and the point at angle −θ is its reflection across the x-axis. Based on this symmetry and the fact that sine is an odd function, what is sin(−θ) in terms of sin(θ)?
Explanation: This question tests understanding of the odd symmetry property of trigonometric functions. Sine is an odd function, meaning sin(-θ) = -sin(θ), which can be visualized on the unit circle where the point at angle -θ is the reflection of the point at angle θ across the x-axis. On the unit circle, a point at angle θ has coordinates (cos(θ), sin(θ)), and the point at angle -θ has coordinates (cos(-θ), sin(-θ)) = (cos(θ), -sin(θ)), reflecting across the x-axis. Choice B is correct because it applies the odd property of sine: sin(-θ) = -sin(θ). Choice A incorrectly treats sine as an even function, using sin(-θ) = sin(θ), when sine is actually odd with sin(-θ) = -sin(θ). Key to symmetry and periodicity: remember that cosine is EVEN (cos(-x) = cos(x)), while sine and tangent are ODD (sin(-x) = -sin(x), tan(-x) = -tan(x)), and that sine and cosine repeat every 2π while tangent repeats every π.
The tangent function repeats every π radians: tan(θ+π)=tan(θ). Using this periodicity, what is tan(45π)?
Explanation: This question tests understanding of the periodicity of trigonometric functions. The sine and cosine functions are periodic with period 2π, meaning sin(θ + 2π) = sin(θ) and cos(θ + 2π) = cos(θ), while tangent has a shorter period of π, meaning tan(θ + π) = tan(θ). Since tangent has period π, we can write 5π/4 = π/4 + π, so tan(5π/4) = tan(π/4) = 1. Choice A is correct because it uses the period correctly to reduce the angle and evaluate at the equivalent special angle. Choice B incorrectly treats tangent as an odd function without considering the specific angle, or confuses with cosine values. For tangent, remember its period is π (not 2π) because tan(θ) = sin(θ)/cos(θ), and both sine and cosine change sign when moving π radians, making their ratio the same. When working with properties, apply them systematically: first reduce using periodicity (add/subtract multiples of the period), then apply even/odd symmetry, then evaluate using special angles or the unit circle.
If sin(6π)=21, what is sin(−6π)? (Use the odd symmetry of sine.)
Explanation: This question tests understanding of the even/odd symmetry properties of trigonometric functions. Sine and tangent are odd functions, meaning sin(-θ) = -sin(θ) and tan(-θ) = -tan(θ), while cosine is an even function, meaning cos(-θ) = cos(θ). For angle π/6, we use the odd property of sine: sin(-π/6) = -sin(π/6) = -1/2. Choice B is correct because it applies the odd property correctly: sin(-π/6) = -sin(π/6) = -1/2. Choice A forgets to apply the negative sign from the odd function property, giving the magnitude correct but the wrong sign. Key to symmetry and periodicity: remember that cosine is EVEN (cos(-x) = cos(x)), while sine and tangent are ODD (sin(-x) = -sin(x), tan(-x) = -tan(x)), and that sine and cosine repeat every 2π while tangent repeats every π. The unit circle provides geometric intuition: reflecting across the x-axis (going from θ to -θ) keeps the x-coordinate (cosine) the same but flips the y-coordinate (sine), explaining why cosine is even and sine is odd.
If sin(6π)=21, then using the odd symmetry of sine, what is sin(−6π)?
Explanation: This question tests understanding of the odd symmetry property of trigonometric functions. Sine is an odd function, meaning sin(-θ) = -sin(θ), which can be verified using the unit circle where reflecting a point across the x-axis changes the sign of the y-coordinate. For angle π/6, we use the odd property of sine: sin(-π/6) = -sin(π/6) = -(1/2) = -1/2. Choice B is correct because it applies the odd property correctly, recognizing that sin(-π/6) = -sin(π/6) = -1/2. Choice A incorrectly gives 1/2, forgetting to apply the negative sign from the odd function property, giving the magnitude correct but the wrong sign. The unit circle provides geometric intuition: reflecting across the x-axis (going from θ to -θ) keeps the x-coordinate (cosine) the same but flips the y-coordinate (sine), explaining why sine is odd.
On the unit circle, adding 2π to an angle corresponds to one full rotation back to the same point. Using this periodicity, what is sin(49π)?
Explanation: This question tests understanding of the periodicity of trigonometric functions. The sine and cosine functions are periodic with period 2π, meaning sin(θ + 2π) = sin(θ) and cos(θ + 2π) = cos(θ), while tangent has a shorter period of π, meaning tan(θ + π) = tan(θ). Since sine has period 2π, we can write 9π/4 = π/4 + 2π, so sin(9π/4) = sin(π/4) = √2/2. Choice A is correct because it uses the period correctly, recognizing that 9π/4 = π/4 + 8π/4 = π/4 + 2π, and sin(π/4) = √2/2. Choice B incorrectly applies a negative sign, perhaps confusing periodicity with the odd property or miscalculating the reference angle. To evaluate trig functions at negative angles or angles beyond 2π, first use periodicity to reduce to the range [0, 2π), then use even/odd properties if the angle is negative, and finally use reference angles if needed.
Using the periodicity property of cosine, cos(θ+2π)=cos(θ), what is cos(49π)?
Explanation: This question tests understanding of the periodicity of trigonometric functions. The sine and cosine functions are periodic with period 2π, meaning sin(θ + 2π) = sin(θ) and cos(θ + 2π) = cos(θ), while tangent has a shorter period of π, meaning tan(θ + π) = tan(θ). Since cosine has period 2π, we can write 9π/4 = π/4 + 2π, so cos(9π/4) = cos(π/4) = √2/2. Choice A is correct because it uses the period correctly. Choice B incorrectly uses the wrong period, claiming the period of cosine is π when it's actually 2π. To evaluate trig functions at angles beyond 2π, first use periodicity to reduce to the range [0, 2π), then use reference angles if needed. When working with properties, apply them systematically: first reduce using periodicity (add/subtract multiples of the period), then apply even/odd symmetry if negative, then evaluate using special angles or the unit circle.
Given the odd-function symmetry of tangent (its graph is symmetric about the origin), what is tan(−θ) in terms of tan(θ)?
Explanation: This question tests understanding of the even/odd symmetry properties of trigonometric functions. Sine and tangent are odd functions, meaning sin(-θ) = -sin(θ) and tan(-θ) = -tan(θ), while cosine is an even function, meaning cos(-θ) = cos(θ). For angle θ, we use the odd property of tangent: tan(-θ) = -tan(θ). Choice B is correct because it applies the odd property correctly. Choice A incorrectly treats tangent as an even function, using tan(-θ) = tan(θ), when tangent is actually odd with tan(-θ) = -tan(θ). Key to symmetry and periodicity: remember that cosine is EVEN (cos(-x) = cos(x)), while sine and tangent are ODD (sin(-x) = -sin(x), tan(-x) = -tan(x)), and that sine and cosine repeat every 2π while tangent repeats every π. The unit circle provides geometric intuition: reflecting across the x-axis (going from θ to -θ) keeps the x-coordinate (cosine) the same but flips the y-coordinate (sine), explaining why cosine is even and sine is odd.
Using the periodicity property of cosine, cos(θ+2π)=cos(θ). What is cos(−47π) expressed using symmetry and/or periodicity as an equivalent standard-angle cosine value?
Explanation: This question tests understanding of both symmetry and periodicity of trigonometric functions. The sine and cosine functions are periodic with period 2π, meaning sin(θ + 2π) = sin(θ) and cos(θ + 2π) = cos(θ), while tangent has a shorter period of π, meaning tan(θ + π) = tan(θ). To evaluate cos(-7π/4), we first use the even property: cos(-7π/4) = cos(7π/4), then periodicity: 7π/4 = -π/4 + 2π, so cos(7π/4) = cos(-π/4) = cos(π/4). Choice A is correct because it combines both properties in proper sequence with specific values. Choice B incorrectly treats cosine as an odd function, giving cos(-θ) = -cos(θ), when cosine is actually even with cos(-θ) = cos(θ). When working with properties, apply them systematically: first reduce using periodicity (add/subtract multiples of the period), then apply even/odd symmetry, then evaluate using special angles or the unit circle. The unit circle provides geometric intuition: reflecting across the x-axis (going from θ to -θ) keeps the x-coordinate (cosine) the same but flips the y-coordinate (sine), explaining why cosine is even and sine is odd.
Using the periodicity property cos(θ+2π)=cos(θ), what is cos(613π)?
Explanation: This question tests understanding of the periodicity of trigonometric functions. The sine and cosine functions are periodic with period 2π, meaning sin(θ + 2π) = sin(θ) and cos(θ + 2π) = cos(θ), while tangent has a shorter period of π, meaning tan(θ + π) = tan(θ). Since cosine has period 2π, we can write 13π/6 = π/6 + 2π, so cos(13π/6) = cos(π/6) = √3/2. Choice B is correct because it uses the period correctly to reduce the angle and evaluate at the equivalent special angle. Choice A forgets to apply the negative sign from the odd function property, giving the magnitude correct but the wrong sign. To evaluate trig functions at negative angles or angles beyond 2π, first use periodicity to reduce to the range [0, 2π), then use even/odd properties if the angle is negative, and finally use reference angles if needed. When working with properties, apply them systematically: first reduce using periodicity (add/subtract multiples of the period), then apply even/odd symmetry, then evaluate using special angles or the unit circle.
Consider the function g(x)=tan(x). If a and b are two values such that g(a)=g(b) and 0<a<b<2π, what is the minimum possible value of b−a?
Explanation: When you encounter questions about periodic functions having equal values, you're exploring the concept of periodicity and how functions repeat their output values. The tangent function has period π, meaning tan(x+π)=tan(x) for all values in its domain. This is different from sine and cosine, which have period 2π. So if g(a)=g(b), the smallest positive difference b−a occurs when b=a+π. However, we must be careful about the domain. The tangent function has vertical asymptotes (is undefined) at x=2π+nπ where n is any integer. Within our interval (0,2π), tangent is undefined at x=2π and x=23π. For any value a in the interval (0,2π), we have tan(a)=tan(a+π), and a+π falls in (π,23π). Similarly, for a in (2π,π), we get a+π in (23π,2π). The minimum difference is π. Choice B (2π) might tempt you if you're thinking about the distance between asymptotes, but this doesn't give equal function values. Choice C (23π) could arise from incorrectly adding half-periods, while choice D (2π) confuses tangent's period with that of sine or cosine. Study tip: Remember that tangent has period π, not 2π like sine and cosine. Always check the specific period of trigonometric functions when solving repetition problems.