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.

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.

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.

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.

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. Choice C is correct because it uses the period correctly. Choice B incorrectly 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. 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.

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.

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 sin(-13π/6), we first use periodicity to simplify: -13π/6 = -π/6 - 2π, so sin(-13π/6) = sin(-π/6). Then we use the odd property: sin(-π/6) = -sin(π/6) = -1/2. Choice B is correct because it combines both properties in proper sequence with specific values. Choice A forgets to apply the negative sign from the odd function property, giving the magnitude correct but the wrong sign. 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.

This question tests understanding of the even/odd symmetry properties of trigonometric functions. Even functions have graphs symmetric about the y-axis (f(-x) = f(x)), while odd functions have graphs symmetric about the origin (f(-x) = -f(x)), and these symmetries are visible in the unit circle geometry. On the unit circle, point (cos(θ), sin(θ)) at angle θ reflects across the x-axis to point (cos(θ), -sin(θ)) at angle -θ, showing why the x-coordinate (cosine) stays the same while the y-coordinate (sine) changes sign. Choice B is correct because it applies the even property correctly. Choice A confuses the symmetry types, claiming cosine has odd symmetry when it actually has even symmetry. 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.

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.

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. Choice C is correct because it applies the odd property correctly. 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. 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.