How to find positive sine - ACT Math

Card 0 of 36

Question

If , what is ? Round to the nearest hundredth.

Answer

Recall that the sine wave is symmetrical with respect to the origin. Therefore, for any value , the value for is . Therefore, if is , then for , it will be .

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Question

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

Answer

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

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Question

Solve for :

if $\frac{-\pi}{2} \le 3x \le \frac{\pi}{2}$

Answer

Recall that the standard triangle, in radians, looks like:

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

Therefore, you can say that must equal $\frac{\pi}{6}$ :

$3x = \frac{\pi}{6}$

Solving for , you get:

$x=\frac{\pi}{18}$

Compare your answer with the correct one above

Question

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

Answer

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:

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

Now, you can treat like it is any standard denominator. Therefore:

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

Combine your fractions and get:

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

Now, from our trig identities, we know that , so we can say:

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

Now, for our triangle, the is $\frac{12}{13}$ . Therefore,

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

Compare your answer with the correct one above

Question

If , what is ? Round to the nearest hundredth.

Answer

Recall that the sine wave is symmetrical with respect to the origin. Therefore, for any value , the value for is . Therefore, if is , then for , it will be .

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

Solve for :

if $\frac{-\pi}{2} \le 3x \le \frac{\pi}{2}$

Answer

Recall that the standard triangle, in radians, looks like:

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

Therefore, you can say that must equal $\frac{\pi}{6}$ :

$3x = \frac{\pi}{6}$

Solving for , you get:

$x=\frac{\pi}{18}$

Compare your answer with the correct one above

Question

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

Answer

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:

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

Now, you can treat like it is any standard denominator. Therefore:

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

Combine your fractions and get:

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

Now, from our trig identities, we know that , so we can say:

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

Now, for our triangle, the is $\frac{12}{13}$ . Therefore,

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

Compare your answer with the correct one above

Question

If , what is ? Round to the nearest hundredth.

Answer

Recall that the sine wave is symmetrical with respect to the origin. Therefore, for any value , the value for is . Therefore, if is , then for , it will be .

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

Solve for :

if $\frac{-\pi}{2} \le 3x \le \frac{\pi}{2}$

Answer

Recall that the standard triangle, in radians, looks like:

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

Therefore, you can say that must equal $\frac{\pi}{6}$ :

$3x = \frac{\pi}{6}$

Solving for , you get:

$x=\frac{\pi}{18}$

Compare your answer with the correct one above

Question

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

Answer

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:

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

Now, you can treat like it is any standard denominator. Therefore:

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

Combine your fractions and get:

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

Now, from our trig identities, we know that , so we can say:

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

Now, for our triangle, the is $\frac{12}{13}$ . Therefore,

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

Compare your answer with the correct one above

Question

If , what is ? Round to the nearest hundredth.

Answer

Recall that the sine wave is symmetrical with respect to the origin. Therefore, for any value , the value for is . Therefore, if is , then for , it will be .

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

Solve for :

if $\frac{-\pi}{2} \le 3x \le \frac{\pi}{2}$

Answer

Recall that the standard triangle, in radians, looks like:

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

Therefore, you can say that must equal $\frac{\pi}{6}$ :

$3x = \frac{\pi}{6}$

Solving for , you get:

$x=\frac{\pi}{18}$

Compare your answer with the correct one above

Question

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

Answer

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:

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

Now, you can treat like it is any standard denominator. Therefore:

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

Combine your fractions and get:

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

Now, from our trig identities, we know that , so we can say:

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

Now, for our triangle, the is $\frac{12}{13}$ . Therefore,

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

Compare your answer with the correct one above

Question

If , what is ? Round to the nearest hundredth.

Answer

Recall that the sine wave is symmetrical with respect to the origin. Therefore, for any value , the value for is . Therefore, if is , then for , it will be .

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

Solve for :

if $\frac{-\pi}{2} \le 3x \le \frac{\pi}{2}$

Answer

Recall that the standard triangle, in radians, looks like:

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

Therefore, you can say that must equal $\frac{\pi}{6}$ :

$3x = \frac{\pi}{6}$

Solving for , you get:

$x=\frac{\pi}{18}$

Compare your answer with the correct one above

Question

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

Answer

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:

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

Now, you can treat like it is any standard denominator. Therefore:

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

Combine your fractions and get:

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

Now, from our trig identities, we know that , so we can say:

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

Now, for our triangle, the is $\frac{12}{13}$ . Therefore,

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

Compare your answer with the correct one above

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