9. Trigonometry

Answer: $\frac{\sqrt{2}}{2}$

$\frac{3\pi}{4}$ is a special angle, so the value is exact rather than a decimal. Read it off the unit circle: $\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}$.

Common mistakes, and the answer each one gives:

• Gave the cosine instead → $- \frac{\sqrt{2}}{2}$
• Worked in degrees, reading the number as an angle in degrees → $\sin{\left(\frac{\pi}{240} \right)}$

Answer: $\frac{\pi}{4}$

Divide by $3$ first: $\cos(x) = \frac{\sqrt{2}}{2}$. That is a special value, and the smallest angle at or above zero with it is $\frac{\pi}{4}$. (Others follow every $2\pi$, and there is a second one inside the first turn unless the value is $1$.)

Common mistakes, and the answer each one gives:

• Gave the other solution inside the first turn, which is larger → $\frac{7 \pi}{4}$
• Forgot to divide by the coefficient before taking the inverse → $0$

Answer: $9 \sin{\left(x \right)}$

Use $\sin^2(x) + \cos^2(x) = 1$ and $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$. Here that reduces the expression to $9 \sin{\left(x \right)}$.

Common mistakes, and the answer each one gives:

• Cancelled to the cosine rather than the sine → $9 \cos{\left(x \right)}$
• Left the tangent untouched → $9 \tan{\left(x \right)}$

Answer: $\frac{1}{2}$

$\frac{\pi}{3}$ is a special angle, so the value is exact rather than a decimal. Read it off the unit circle: $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$.

Common mistakes, and the answer each one gives:

• Gave the sine instead → $\frac{\sqrt{3}}{2}$
• Worked in degrees, reading the number as an angle in degrees → $\cos{\left(\frac{\pi}{540} \right)}$

Answer: $\frac{\pi}{4}$

Divide by $4$ first: $\sin(x) = \frac{\sqrt{2}}{2}$. That is a special value, and the smallest angle at or above zero with it is $\frac{\pi}{4}$. (Others follow every $2\pi$, and there is a second one inside the first turn unless the value is $1$.)

Common mistakes, and the answer each one gives:

• Gave the other solution inside the first turn, which is larger → $\frac{3 \pi}{4}$
• Forgot to divide by the coefficient before taking the inverse → $\frac{\pi}{2}$

Answer: $5 \sin{\left(x \right)}$

Use $\sin^2(x) + \cos^2(x) = 1$ and $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$. Here that reduces the expression to $5 \sin{\left(x \right)}$.

Common mistakes, and the answer each one gives:

• Cancelled to the cosine rather than the sine → $5 \cos{\left(x \right)}$
• Left the tangent untouched → $5 \tan{\left(x \right)}$

Answer: $- \frac{\sqrt{2}}{2}$

$\frac{3\pi}{4}$ is a special angle, so the value is exact rather than a decimal. Read it off the unit circle: $\cos\left(\frac{3\pi}{4}\right) = - \frac{\sqrt{2}}{2}$.

Common mistakes, and the answer each one gives:

• Gave the sine instead → $\frac{\sqrt{2}}{2}$
• Worked in degrees, reading the number as an angle in degrees → $\cos{\left(\frac{\pi}{240} \right)}$

Answer: $\frac{\pi}{6}$

Divide by $2$ first: $\sin(x) = \frac{1}{2}$. That is a special value, and the smallest angle at or above zero with it is $\frac{\pi}{6}$. (Others follow every $2\pi$, and there is a second one inside the first turn unless the value is $1$.)

Common mistakes, and the answer each one gives:

• Gave the other solution inside the first turn, which is larger → $\frac{5 \pi}{6}$
• Forgot to divide by the coefficient before taking the inverse → $\frac{\pi}{2}$

Answer: $6$

Use $\sin^2(x) + \cos^2(x) = 1$ and $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$. Here that reduces the expression to $6$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of using the identity → $12$
• Treated $\sin^2 + \cos^2$ as $0$ → $0$

Answer: $\frac{\sqrt{3}}{2}$

$\frac{\pi}{6}$ is a special angle, so the value is exact rather than a decimal. Read it off the unit circle: $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$.

Common mistakes, and the answer each one gives:

• Gave the sine instead → $\frac{1}{2}$
• Worked in degrees, reading the number as an angle in degrees → $\cos{\left(\frac{\pi}{1080} \right)}$

Answer: $\frac{\pi}{4}$

Divide by $2$ first: $\sin(x) = \frac{\sqrt{2}}{2}$. That is a special value, and the smallest angle at or above zero with it is $\frac{\pi}{4}$. (Others follow every $2\pi$, and there is a second one inside the first turn unless the value is $1$.)

Common mistakes, and the answer each one gives:

• Gave the other solution inside the first turn, which is larger → $\frac{3 \pi}{4}$
• Forgot to divide by the coefficient before taking the inverse → $\frac{\pi}{2}$

Answer: $3 \sin{\left(x \right)}$

Use $\sin^2(x) + \cos^2(x) = 1$ and $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$. Here that reduces the expression to $3 \sin{\left(x \right)}$.

Common mistakes, and the answer each one gives:

• Cancelled to the cosine rather than the sine → $3 \cos{\left(x \right)}$
• Left the tangent untouched → $3 \tan{\left(x \right)}$

Answer: $\frac{1}{2}$

$\frac{5\pi}{6}$ is a special angle, so the value is exact rather than a decimal. Read it off the unit circle: $\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}$.

Common mistakes, and the answer each one gives:

• Gave the cosine instead → $- \frac{\sqrt{3}}{2}$
• Worked in degrees, reading the number as an angle in degrees → $\sin{\left(\frac{\pi}{216} \right)}$

Answer: $\frac{\pi}{3}$

Divide by $7$ first: $\sin(x) = \frac{\sqrt{3}}{2}$. That is a special value, and the smallest angle at or above zero with it is $\frac{\pi}{3}$. (Others follow every $2\pi$, and there is a second one inside the first turn unless the value is $1$.)

Common mistakes, and the answer each one gives:

• Gave the other solution inside the first turn, which is larger → $\frac{2 \pi}{3}$
• Forgot to divide by the coefficient before taking the inverse → $\frac{\pi}{2}$

Answer: $9$

Use $\sin^2(x) + \cos^2(x) = 1$ and $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$. Here that reduces the expression to $9$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of using the identity → $18$
• Treated $\sin^2 + \cos^2$ as $0$ → $0$

Answer: $0$

$\pi$ is a special angle, so the value is exact rather than a decimal. Read it off the unit circle: $\sin\left(\pi\right) = 0$.

Common mistakes, and the answer each one gives:

• Gave the cosine instead → $-1$
• Worked in degrees, reading the number as an angle in degrees → $\sin{\left(\frac{\pi}{180} \right)}$

Answer: $\frac{\pi}{4}$

Divide by $9$ first: $\sin(x) = \frac{\sqrt{2}}{2}$. That is a special value, and the smallest angle at or above zero with it is $\frac{\pi}{4}$. (Others follow every $2\pi$, and there is a second one inside the first turn unless the value is $1$.)

Common mistakes, and the answer each one gives:

• Gave the other solution inside the first turn, which is larger → $\frac{3 \pi}{4}$
• Forgot to divide by the coefficient before taking the inverse → $\frac{\pi}{2}$

Answer: $4$

Use $\sin^2(x) + \cos^2(x) = 1$ and $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$. Here that reduces the expression to $4$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of using the identity → $8$
• Treated $\sin^2 + \cos^2$ as $0$ → $0$

Answer: $\frac{\sqrt{3}}{2}$

$\frac{\pi}{3}$ is a special angle, so the value is exact rather than a decimal. Read it off the unit circle: $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$.

Common mistakes, and the answer each one gives:

• Gave the cosine instead → $\frac{1}{2}$
• Worked in degrees, reading the number as an angle in degrees → $\sin{\left(\frac{\pi}{540} \right)}$

Answer: $\frac{\pi}{3}$

Divide by $6$ first: $\cos(x) = \frac{1}{2}$. That is a special value, and the smallest angle at or above zero with it is $\frac{\pi}{3}$. (Others follow every $2\pi$, and there is a second one inside the first turn unless the value is $1$.)

Common mistakes, and the answer each one gives:

• Gave the other solution inside the first turn, which is larger → $\frac{5 \pi}{3}$
• Forgot to divide by the coefficient before taking the inverse → $0$

Answer: $7 \sin{\left(x \right)}$

Use $\sin^2(x) + \cos^2(x) = 1$ and $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$. Here that reduces the expression to $7 \sin{\left(x \right)}$.

Common mistakes, and the answer each one gives:

• Cancelled to the cosine rather than the sine → $7 \cos{\left(x \right)}$
• Left the tangent untouched → $7 \tan{\left(x \right)}$

Answer: $0$

$\frac{\pi}{2}$ is a special angle, so the value is exact rather than a decimal. Read it off the unit circle: $\cos\left(\frac{\pi}{2}\right) = 0$.

Common mistakes, and the answer each one gives:

• Gave the sine instead → $1$
• Worked in degrees, reading the number as an angle in degrees → $\cos{\left(\frac{\pi}{360} \right)}$

Answer: $\frac{\pi}{4}$

Divide by $5$ first: $\cos(x) = \frac{\sqrt{2}}{2}$. That is a special value, and the smallest angle at or above zero with it is $\frac{\pi}{4}$. (Others follow every $2\pi$, and there is a second one inside the first turn unless the value is $1$.)

Common mistakes, and the answer each one gives:

• Gave the other solution inside the first turn, which is larger → $\frac{7 \pi}{4}$
• Forgot to divide by the coefficient before taking the inverse → $0$

Answer: $8 \sin{\left(x \right)}$

Use $\sin^2(x) + \cos^2(x) = 1$ and $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$. Here that reduces the expression to $8 \sin{\left(x \right)}$.

Common mistakes, and the answer each one gives:

• Cancelled to the cosine rather than the sine → $8 \cos{\left(x \right)}$
• Left the tangent untouched → $8 \tan{\left(x \right)}$

Answer: $- \frac{\sqrt{3}}{2}$

$\frac{5\pi}{6}$ is a special angle, so the value is exact rather than a decimal. Read it off the unit circle: $\cos\left(\frac{5\pi}{6}\right) = - \frac{\sqrt{3}}{2}$.

Common mistakes, and the answer each one gives:

• Gave the sine instead → $\frac{1}{2}$
• Worked in degrees, reading the number as an angle in degrees → $\cos{\left(\frac{\pi}{216} \right)}$

Answer: $\frac{\pi}{4}$

Divide by $8$ first: $\cos(x) = \frac{\sqrt{2}}{2}$. That is a special value, and the smallest angle at or above zero with it is $\frac{\pi}{4}$. (Others follow every $2\pi$, and there is a second one inside the first turn unless the value is $1$.)

Common mistakes, and the answer each one gives:

• Gave the other solution inside the first turn, which is larger → $\frac{7 \pi}{4}$
• Forgot to divide by the coefficient before taking the inverse → $0$

Answer: $7$

Use $\sin^2(x) + \cos^2(x) = 1$ and $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$. Here that reduces the expression to $7$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of using the identity → $14$
• Treated $\sin^2 + \cos^2$ as $0$ → $0$

Answer: $-1$

$\pi$ is a special angle, so the value is exact rather than a decimal. Read it off the unit circle: $\cos\left(\pi\right) = -1$.

Common mistakes, and the answer each one gives:

• Gave the sine instead → $0$
• Worked in degrees, reading the number as an angle in degrees → $\cos{\left(\frac{\pi}{180} \right)}$

Answer: $\frac{\pi}{6}$

Divide by $8$ first: $\sin(x) = \frac{1}{2}$. That is a special value, and the smallest angle at or above zero with it is $\frac{\pi}{6}$. (Others follow every $2\pi$, and there is a second one inside the first turn unless the value is $1$.)

Common mistakes, and the answer each one gives:

• Gave the other solution inside the first turn, which is larger → $\frac{5 \pi}{6}$
• Forgot to divide by the coefficient before taking the inverse → $\frac{\pi}{2}$

Answer: $5$

Use $\sin^2(x) + \cos^2(x) = 1$ and $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$. Here that reduces the expression to $5$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of using the identity → $10$
• Treated $\sin^2 + \cos^2$ as $0$ → $0$