Drill 1 · Math · Sine and Cosine Function Values
AP Precalculus: Sine and Cosine Function Values (Drill 22) is a Math practice drill covering Sine and Cosine Function Values. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.
This AP(r) Precalculus drill covers sine and cosine as functions (Topics 3.3-3.4): domain, range, even and odd properties, the graphs of y = sin x and y = cos x, and evaluating or interpreting sine and cosine values from graphs and equations. These functions appear throughout the rest of Unit 3.
Question 1. Which of the following statements is true for all real numbers \( x \)?
Explanation: Choice D is correct. Cosine is an even function: \( \cos(-x) = \cos(x) \) for all \( x \). This follows from the unit circle -- the point corresponding to angle \( -x \) is the reflection of the point for \( x \) across the x-axis, so the x-coordinate (which equals cosine) is unchanged. Choice A is incorrect because sine is an odd function, not even: \( \sin(-x) = -\sin(x) \), not \( \sin(x) \). Choice B is incorrect because it states \( \cos(-x) = -\cos(x) \), which would make cosine an odd function. Cosine is even, so the negative sign is wrong. Choice C is incorrect because \( \sin(-x) = -\sin(x) \) in general, not \( \cos(x) \). This confuses the relationship between sine and cosine as a phase shift, which applies only for specific arguments (e.g., \( \sin(x) = \cos(\frac{\pi}{2} - x) \)), not generally for \( -x \).
Question 2. Which of the following statements about \( y = \cos x \) is FALSE?
Explanation: Choice B is correct (it is the false statement). The function \( y = \cos x \) is DECREASING, not increasing, on the interval \( (0, \pi) \). At \( x=0 \), \( \cos(0) = 1 \) (its maximum), and it decreases steadily to \( \cos(\pi) = -1 \) (its minimum). Choice A is incorrect (it is a true statement): the period of \( y = \cos x \) is \( 2\pi \) -- the function completes one full cycle every \( 2\pi \) units. Choice C is incorrect (it is a true statement): the amplitude of \( y = \cos x \) is 1 -- the function oscillates between -1 and 1, giving a maximum displacement of 1 from the midline. Choice D is incorrect (it is a true statement): at \( x=0 \), \( \cos(0) = 1 \), so the graph does pass through \( (0, 1) \).
Question 3. If \( \sin x = \dfrac{3}{5} \) and \( x \) is in the first quadrant, what is the value of \( \cos x \)?
Explanation: Choice A is correct. Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \): \( \cos^2 x = 1 - \sin^2 x = 1 - \dfrac{9}{25} = \dfrac{16}{25} \). So \( \cos x = \pm \dfrac{4}{5} \). Since \( x \) is in the first quadrant, cosine is positive: \( \cos x = \dfrac{4}{5} \). Choice B is incorrect because \( \frac{3}{4} \) comes from dividing \( \sin x \) by \( \cos x \) (computing tangent instead of cosine), or from using \( \cos x = 1 - \sin x = 1 - \frac{3}{5} = \frac{2}{5} \) with an arithmetic error. Choice C is incorrect because the magnitude \( \frac{4}{5} \) is correct, but the negative sign is wrong -- in the first quadrant, both sine and cosine are positive. This would be correct if \( x \) were in the second quadrant. Choice D is incorrect because \( \frac{5}{4} \) exceeds 1, which is impossible since \( -1 \leq \cos x \leq 1 \) for all \( x \). This likely comes from inverting the fraction: taking \( \frac{5}{4} \) instead of \( \frac{4}{5} \).
Question 4. A 5-meter ladder leans against a wall. The angle \( \theta \) between the ladder and the ground satisfies \( \sin \theta = \dfrac{4}{5} \). How high up the wall does the ladder reach?
Explanation: Choice C is correct. The height \( h \) reached by the ladder satisfies \( h = 5 \cdot \sin \theta = 5 \cdot \dfrac{4}{5} = 4 \) meters. The ladder reaches 4 meters up the wall. Choice A is incorrect because 3.2 meters likely comes from computing \( 5 \times \frac{4}{5} = 4 \) incorrectly, perhaps canceling incorrectly to get \( \frac{4 \times 5}{5 \times 5} = \frac{20}{25} = 0.8 \times 4 = 3.2 \). Choice B is incorrect because 3 meters would be the correct answer if the ladder's base-to-wall distance were requested (using \( \cos \theta \)): \( \cos \theta = \frac{3}{5} \) gives \( 5 \cdot \frac{3}{5} = 3 \) meters -- but that is the horizontal distance, not the height. Choice D is incorrect because 5 meters is the full length of the ladder, not the height it reaches. The ladder would reach 5 meters only if it were perfectly vertical (\( \theta = 90\circ \)), but here \( \sin \theta = \frac{4}{5} < 1 \).
Question 5. Let \( f(x) = \sin x \). Which of the following statements about \( f \) is true?
Explanation: Choice D is correct. The average rate of change of \( \sin x \) on \( [0, \frac{\pi}{2}] \) is \( \dfrac{f(\frac{\pi}{2}) - f(0)}{\frac{\pi}{2} - 0} = \dfrac{1 - 0}{\frac{\pi}{2}} = \dfrac{2}{\pi} \). Choice A is incorrect because \( \sin\!\left(\frac{\pi}{2}\right) = 1 \), not 0. The zeros of \( \sin x \) occur at integer multiples of \( \pi \) (i.e., \( x = 0, \pm\pi, \pm 2\pi, \ldots \)), not at \( \frac{\pi}{2} \). Choice B is incorrect because \( \sin x \) is an ODD function: \( \sin(-x) = -\sin(x) \) for all \( x \). An even function would satisfy \( f(-x) = f(x) \), which sine does not. Choice C is incorrect because \( \sin(x + \pi) = -\sin(x) \), not \( \sin(x) \). The period of sine is \( 2\pi \), not \( \pi \): shifting by \( \pi \) reflects the function, giving the negative.