ACT Math : How to find positive sine

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #1 : How To Find Positive Sine

If \(\displaystyle sin(x)=0.41\), what is \(\displaystyle sin(-x)\)?  Round to the nearest hundredth.

Possible Answers:

\(\displaystyle -0.82\)

\(\displaystyle 0.82\)

\(\displaystyle 0.41\)

\(\displaystyle 0.54\)

\(\displaystyle -0.41\)

Correct answer:

\(\displaystyle -0.41\)

Explanation:

Recall that the sine wave is symmetrical with respect to the origin. Therefore, for any value \(\displaystyle (x,y)\), the value for \(\displaystyle -x\) is \(\displaystyle -y\). Therefore, if \(\displaystyle sin(x)\) is \(\displaystyle 0.41\), then for \(\displaystyle -x\), it will be \(\displaystyle -0.41\).

Example Question #1 : How To Find Positive Sine

In a right triangle, cos(A) = \(\displaystyle \frac{11}{14}\). What is sin(A)?

Possible Answers:

\(\displaystyle 5\sqrt{3}\)

\(\displaystyle \frac{11}{75}\)

\(\displaystyle \frac{5\sqrt{3}}{11}\)

\(\displaystyle \frac{5\sqrt{3}}{14}\)

\(\displaystyle \frac{75}{14}\)

Correct answer:

\(\displaystyle \frac{5\sqrt{3}}{14}\)

Explanation:

In a right triangle, for sides a and b, with c being the hypotenuse, \(\displaystyle a^{2} + b ^{2} = c^{2}\). Thus if cos(A) is \(\displaystyle \frac{11}{14}\), then c = 14, and the side adjacent to A is 11. Therefore, the side opposite of angle A is the square root of \(\displaystyle 14^{2} - 11^{2}\), which is \(\displaystyle \sqrt{75} = 5\sqrt{3}.\) Since sin is \(\displaystyle \frac{opposite}{hypotenuse}\), sin(A) is \(\displaystyle \frac{5\sqrt{3}}{14}\).

Example Question #2985 : Act Math

51213

What is the value of \(\displaystyle cos(x)(\frac{cos(x)}{\sin(x)})+sin(x)\)?

Possible Answers:

\(\displaystyle 1\)

\(\displaystyle \frac{12}{13}\)

\(\displaystyle \frac{15}{7}\)

\(\displaystyle \frac{1}{2}\)

\(\displaystyle \frac{13}{12}\)

Correct answer:

\(\displaystyle \frac{13}{12}\)

Explanation:

As with all trigonometry problems, begin by considering how you could rearrange the question. They often have hidden easy ways out. So begin by noticing:

\(\displaystyle cos(x)(\frac{cos(x)}{\sin(x)})+sin(x)=(\frac{cos^2(x)}{\sin(x)})+sin(x)\)

Now, you can treat \(\displaystyle sin(x)\) like it is any standard denominator. Therefore:

\(\displaystyle (\frac{cos^2(x)}{\sin(x)})+sin(x)=(\frac{cos^2(x)}{\sin(x)})+\frac{sin^2(x)}{sin(x)}\)

Combine your fractions and get:

\(\displaystyle \frac{cos^2(x)+ sin^2(x)}{sin(x)}\)

Now, from our trig identities, we know that \(\displaystyle cos^2(x)+ sin^2(x)=1\), so we can say:

\(\displaystyle \frac{cos^2(x)+ sin^2(x)}{sin(x)}=\frac{1}{sin(x)}\)

Now, for our triangle, the \(\displaystyle sin(x)\) is \(\displaystyle \frac{12}{13}\). Therefore,

\(\displaystyle \frac{1}{sin(x)} = \frac{1}{\frac{12}{13}}=\frac{13}{12}\)

Example Question #1 : How To Find Positive Sine

Solve for \(\displaystyle x\):

\(\displaystyle sin(3x)=0.5\) if \(\displaystyle \frac{-\pi}{2} \le 3x \le \frac{\pi}{2}\)

Possible Answers:

\(\displaystyle \frac{\pi}{18}\)

\(\displaystyle \frac{\pi}{6}\)

\(\displaystyle 3\pi\)

\(\displaystyle \frac{3\pi}{2}\)

\(\displaystyle \frac{\pi}{2}\)

Correct answer:

\(\displaystyle \frac{\pi}{18}\)

Explanation:

\(\displaystyle sin(3x)=0.5\)

Recall that the standard \(\displaystyle 30-60-90\) triangle, in radians, looks like:

Rt1

Since \(\displaystyle sin(x) = \frac{opposite}{hypotenuse}\), you can tell that \(\displaystyle sin(\frac{\pi}{6}) = \frac{1}{2}\).

Therefore, you can say that \(\displaystyle 3x\) must equal \(\displaystyle \frac{\pi}{6}\):

\(\displaystyle 3x = \frac{\pi}{6}\)

Solving for \(\displaystyle x\), you get:

\(\displaystyle x=\frac{\pi}{18}\)

 

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