What are the six trig function values of $\frac{π}{3}$ $pi/3$ ?

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What are the six trig function values of $\frac{π}{3}$ $pi/3$ ?

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Answer 1 · 2015-11-26T01:13:31

The $6$ $6$ trigonometric values of $\frac{π}{3}$ $pi/3$ are:

$sin (\frac{π}{3}) = \frac{\sqrt{3}}{2}$ $sin(pi/3)=sqrt(3)/2$
$cos (\frac{π}{3}) = \frac{1}{2}$ $cos(pi/3)=1/2$
$tan (\frac{π}{3}) = \sqrt{3}$ $tan(pi/3)=sqrt(3)$
$csc (\frac{π}{3}) = \frac{2 \sqrt{3}}{3}$ $csc(pi/3)=(2sqrt(3))/3$
$sec (\frac{π}{3}) = 2$ $sec(pi/3)=2$
$cot (\frac{π}{3}) = \frac{\sqrt{3}}{3}$ $cot(pi/3)=sqrt(3)/3$

Explanation:

The $6$ $6$ trigonometric ratios are:

$1 . sin θ$ $1. sintheta$
$2 . cos θ$ $2. costheta$
$3 . tan θ$ $3. tantheta$
$4 . csc θ$ $4. csctheta$
$5 . sec θ$ $5. sectheta$
$6 . cot θ$ $6. cottheta$

Using the ratios, we can determine their values when $θ$ $theta$ is $\frac{π}{3}$ $pi/3$ :

Recall that $π$ $pi$ radians is $180^{\circ}$ $180^@$ .

$1 . sin θ$ $1. sintheta$

$= sin (\frac{π}{3})$ $=sin(pi/3)$

$= sin (\frac{180^{\circ}}{3})$ $=sin(180^@/3)$

$= sin (60^{\circ})$ $=sin(60^@)$

$= \frac{\sqrt{3}}{2}$ $=sqrt(3)/2$

$2 . cos θ$ $2. costheta$

$= cos (\frac{π}{3})$ $=cos(pi/3)$

$= cos (\frac{180^{\circ}}{3})$ $=cos(180^@/3)$

$= cos (60^{\circ})$ $=cos(60^@)$

$= \frac{1}{2}$ $=1/2$

$3 . tan θ$ $3. tantheta$

$= tan (\frac{π}{3})$ $=tan(pi/3)$

$= tan (\frac{180^{\circ}}{3})$ $=tan(180^@/3)$

$= tan (60^{\circ})$ $=tan(60^@)$

$= \frac{\sqrt{3}}{1}$ $=sqrt(3)/1$

$= \sqrt{3}$ $=sqrt(3)$

$4 . csc θ$ $4. csctheta$

$= \frac{1}{sin θ}$ $=1/sintheta$

$= \frac{1}{sin (\frac{π}{3})}$ $=1/sin(pi/3)$

$= \frac{1}{sin (\frac{180^{\circ}}{3})}$ $=1/sin(180^@/3)$

$= \frac{1}{sin (60^{\circ})}$ $=1/sin(60^@)$

$= \frac{1}{\frac{\sqrt{3}}{2}}$ $=1/(sqrt(3)/2)$

$= \frac{2}{\sqrt{3}}$ $=2/sqrt(3)$

$= \frac{2 \sqrt{3}}{3}$ $=(2sqrt(3))/3$

$5 . sec θ$ $5. sectheta$

$= \frac{1}{cos θ}$ $=1/costheta$

$= \frac{1}{cos (\frac{π}{3})}$ $=1/cos(pi/3)$

$= \frac{1}{cos (\frac{180^{\circ}}{3})}$ $=1/cos(180^@/3)$

$= \frac{1}{cos (60^{\circ})}$ $=1/cos(60^@)$

$= \frac{1}{\frac{1}{2}}$ $=1/(1/2)$

$= 2$ $=2$

$6 . cot θ$ $6. cottheta$

$= \frac{1}{tan θ}$ $=1/tantheta$

$= \frac{1}{tan (\frac{π}{3})}$ $=1/tan(pi/3)$

$= \frac{1}{tan (\frac{180^{\circ}}{3})}$ $=1/tan(180^@/3)$

$= \frac{1}{tan (60^{\circ})}$ $=1/tan(60^@)$

$= \frac{1}{\frac{\sqrt{3}}{1}}$ $=1/(sqrt(3)/1)$

$= \frac{1}{\sqrt{3}}$ $=1/sqrt(3)$

$= \frac{\sqrt{3}}{3}$ $=sqrt(3)/3$

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