Question 1

Simplify the given expression algebraically.
$$\cos \left( \frac { \pi } { 2 } + x \right)$$

Accepted Answer

A)  $\sin x$ 
B)  $- \sin x$ 
C)  $\cos x$ 
D)  $- \cos x$ 
E)  1 
A)  $\sin x$ 
B)  $- \sin x$ 
C)  $\cos x$ 
D)  $- \cos x$ 
E)  1

Question 2

Solve the multi-angle equation below.
$\cos \left( \frac { x } { 2 } \right) = - \frac { \sqrt { 3 } } { 2 }$

Accepted Answer

A)  $\frac { 10 \pi } { 3 } + n \pi , \frac { 14 \pi } { 3 } + n \pi$, where $n$ is an integer 
B)  $\frac { 10 \pi } { 3 } + 2 n \pi , \frac { 14 \pi } { 3 } + 2 n \pi$, where $n$ is an integer 
C)  $\frac { 5 \pi } { 3 } + n \pi , \frac { 14 \pi } { 3 } + n \pi$, where $n$ is an integer 
D)  $\frac { 5 \pi } { 3 } + n \pi , \frac { 7 \pi } { 3 } + n \pi$, where $n$ is an integer 
E)  $\frac { 5 \pi } { 3 } + 4 n \pi , \frac { 7 \pi } { 3 } + 4 n \pi$, where $n$ is an integer 
A)  $\frac { 10 \pi } { 3 } + n \pi , \frac { 14 \pi } { 3 } + n \pi$, where $n$ is an integer 
B)  $\frac { 10 \pi } { 3 } + 2 n \pi , \frac { 14 \pi } { 3 } + 2 n \pi$, where $n$ is an integer 
C)  $\frac { 5 \pi } { 3 } + n \pi , \frac { 14 \pi } { 3 } + n \pi$, where $n$ is an integer 
D)  $\frac { 5 \pi } { 3 } + n \pi , \frac { 7 \pi } { 3 } + n \pi$, where $n$ is an integer 
E)  $\frac { 5 \pi } { 3 } + 4 n \pi , \frac { 7 \pi } { 3 } + 4 n \pi$, where $n$ is an integer

Question 3

Use the half-angle formulas to determine the exact value of the following.
$$\cos \left( - 22.5 ^ { \circ } \right)$$

Accepted Answer

A)  $- \frac { \sqrt { 2 + \sqrt { 2 } } } { 2 }$ 
B)  $\frac { \sqrt { 2 - \sqrt { 3 } } } { 2 }$ 
C)  $- \frac { \sqrt { 2 - \sqrt { 3 } } } { 2 }$ 
D)  $\frac { \sqrt { 3 - \sqrt { 2 } } } { 2 }$ 
E)  $\frac { \sqrt { 2 + \sqrt { 3 } } } { 2 }$ 
A)  $- \frac { \sqrt { 2 + \sqrt { 2 } } } { 2 }$ 
B)  $\frac { \sqrt { 2 - \sqrt { 3 } } } { 2 }$ 
C)  $- \frac { \sqrt { 2 - \sqrt { 3 } } } { 2 }$ 
D)  $\frac { \sqrt { 3 - \sqrt { 2 } } } { 2 }$ 
E)  $\frac { \sqrt { 2 + \sqrt { 3 } } } { 2 }$

Question 4

Verify the identity shown below.
$$\frac { 1 - \sin \theta } { 1 + \sin \theta } = 2 \sec ^ { 2 } \theta - 2 \sec \theta \tan \theta - 1$$

Accepted Answer

\[\begin{aligned}
\frac { 1 - \sin \thet

Question 5

If $x = 2 \cot \theta$, use trigonometric substitution to write $\sqrt { 4 + x ^ { 2 } }$ as a trigonometric function of $\theta$, where $0 < \theta < \pi$.

Accepted Answer

A)  $2 \cos \theta$ 
B)  $2 \csc \theta$ 
C)  $2 \cot \theta$ 
D)  $2 \sec \theta$ 
E)  $2 \sin \theta$ 
A)  $2 \cos \theta$ 
B)  $2 \csc \theta$ 
C)  $2 \cot \theta$ 
D)  $2 \sec \theta$ 
E)  $2 \sin \theta$

Question 6

Verify the identity shown below.
$$\frac { \tan \theta + 1 } { \sec \theta + \csc \theta } = \sin \theta$$

Accepted Answer

\[\begin{aligned}
\frac { \tan \theta +

Question 7

Find all solutions of the following equation in the interval $[ 0,2 \pi )$.
$2 \cos x - \sec x = 0$

Accepted Answer

A)  $x = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
B)  $x = \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
C)  $x = \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
D)  $x = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$ 
E)  $x = \frac { \pi } { 3 } , \frac { 3 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 6 }$ 
A)  $x = \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
B)  $x = \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } , \frac { 7 \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
C)  $x = \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
D)  $x = 0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$ 
E)  $x = \frac { \pi } { 3 } , \frac { 3 \pi } { 4 } , \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 6 }$

Question 8

Verify the identity shown below.
$$\csc \theta - \cos \theta \cot \theta = \sin \theta$$

Accepted Answer

None

Question 9

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent.
$\cot \beta \sec \beta$

Accepted Answer

A)  $\frac { 1 } { \sin \beta }$ 
B)  $\frac { \sec \beta } { \tan \beta }$ 
C)  $\frac { 1 } { \cos \beta \tan \beta }$ 
D)  $\sec \beta$ 
E)  $\csc \beta$ 
A)  $\frac { 1 } { \sin \beta }$ 
B)  $\frac { \sec \beta } { \tan \beta }$ 
C)  $\frac { 1 } { \cos \beta \tan \beta }$ 
D)  $\sec \beta$ 
E)  $\csc \beta$

Question 10

Use the sum-to-product formulas to write the given expression as a product. $\sin 9 \theta - \sin 7 \theta$

Accepted Answer

A)  $2 \sin 8 \theta \cos \theta$ 
B)  $2 \cos 8 \theta \cos \theta$ 
C)  $- 2 \sin 8 \theta \sin \theta$ 
D)  $- 2 \cos 8 \theta \cos \theta$ 
E)  $2 \cos 8 \theta \sin \theta$ 
A)  $2 \sin 8 \theta \cos \theta$ 
B)  $2 \cos 8 \theta \cos \theta$ 
C)  $- 2 \sin 8 \theta \sin \theta$ 
D)  $- 2 \cos 8 \theta \cos \theta$ 
E)  $2 \cos 8 \theta \sin \theta$

Question 11

Find the exact value of the given expression using a sum or difference formula. $\sin 345 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
B)  $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
C)  $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
D)  $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
A)  $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
B)  $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
C)  $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
D)  $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$

Question 12

Find the exact value of the given expression using a sum or difference formula. $\sin 165 ^ { \circ }$

Accepted Answer

A)  $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
B)  $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
C)  $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
D)  $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
A)  $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
B)  $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
C)  $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
D)  $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$

Question 13

Determine which of the following are trigonometric identities.
I. $\csc ( \theta ) \sec ( \theta ) = \tan ( \theta )$
II. $\csc ( \theta ) \tan ( \theta ) = \sec ( \theta )$
III. $\tan ( \theta ) \sec ( \theta ) = \csc ( \theta )$
IV. $\csc ( \theta ) \sin ( \theta ) = 1$

Accepted Answer

A)  II is the only identity. 
B)  II and IV are the only identities. 
C)  I, III, and IV are the only identities. 
D)  I is the only identity. 
E)  II, III, and IV are the only identities. 
A)  II is the only identity. 
B)  II and IV are the only identities. 
C)  I, III, and IV are the only identities. 
D)  I is the only identity. 
E)  II, III, and IV are the only identities.

Question 14

Write the given expression as algebraic expression.
$\cos ( \arccos x - \arcsin x )$

Accepted Answer

A)  1 
B)  $\frac { x \sqrt { 1 - x ^ { 2 } } - x } { \sqrt { x ^ { 2 } + 1 } }$ 
C)  0 
D)  $2 x \sqrt { 1 - x ^ { 2 } }$ 
E)  $\frac { x \sqrt { 1 - x ^ { 2 } } + x } { \sqrt { x ^ { 2 } + 1 } }$ 
A)  1 
B)  $\frac { x \sqrt { 1 - x ^ { 2 } } - x } { \sqrt { x ^ { 2 } + 1 } }$ 
C)  0 
D)  $2 x \sqrt { 1 - x ^ { 2 } }$ 
E)  $\frac { x \sqrt { 1 - x ^ { 2 } } + x } { \sqrt { x ^ { 2 } + 1 } }$

Question 15

Find the exact value of $\cos ( u - v )$ given that $\sin u = - \frac { 9 } { 41 }$ and $\cos v = \frac { 15 } { 17 }$. (Both $u$ and $v$ are in Quadrant IV.)

Accepted Answer

A)  $\quad \cos ( u - v ) = - \frac { 194 } { 697 }$ 
B)  $\quad \cos ( u - v ) = - \frac { 39 } { 697 }$ 
C)  $\quad \cos ( u - v ) = \frac { 568 } { 697 }$ 
D)  $\quad \cos ( u - v ) = - \frac { 528 } { 697 }$ 
E)  $\quad \cos ( u - v ) = \frac { 672 } { 697 }$ 
A)  $\quad \cos ( u - v ) = - \frac { 194 } { 697 }$ 
B)  $\quad \cos ( u - v ) = - \frac { 39 } { 697 }$ 
C)  $\quad \cos ( u - v ) = \frac { 568 } { 697 }$ 
D)  $\quad \cos ( u - v ) = - \frac { 528 } { 697 }$ 
E)  $\quad \cos ( u - v ) = \frac { 672 } { 697 }$

Question 16

Write the given expression as the sine of an angle.
$\sin 85 ^ { \circ } \cos 50 ^ { \circ } + \sin 50 ^ { \circ } \cos 85 ^ { \circ }$

Accepted Answer

A)  $\sin \left( - 100 ^ { \circ } \right)$ 
B)  $\sin \left( 135 ^ { \circ } \right)$ 
C)  $\sin \left( 35 ^ { \circ } \right)$ 
D)  $\sin \left( 85 ^ { \circ } \right)$ 
E)  $\sin \left( 50 ^ { \circ } \right)$ 
A)  $\sin \left( - 100 ^ { \circ } \right)$ 
B)  $\sin \left( 135 ^ { \circ } \right)$ 
C)  $\sin \left( 35 ^ { \circ } \right)$ 
D)  $\sin \left( 85 ^ { \circ } \right)$ 
E)  $\sin \left( 50 ^ { \circ } \right)$

Question 17

Verify the given identity.
$$\frac { \cos u - \cos v } { \cos u + \cos v } = - \tan \frac { 1 } { 2 } ( u + v ) \tan \frac { 1 } { 2 } ( u - v )$$

Accepted Answer

None

Question 18

Verify the given identity.
$$\cos ( x + y ) \cos ( x - y ) = \cos ^ { 2 } x - \sin ^ { 2 } y$$

Accepted Answer

\[\begin{aligned}
\cos ( x + y ) \cos (

Question 19

Simplify the given expression algebraically.
$$\cos ( \pi + x )$$

Accepted Answer

A)  $\sin x$ 
B)  $- \sin x$ 
C)  $\cos x$ 
D)  $- \cos x$ 
E)  1 
A)  $\sin x$ 
B)  $- \sin x$ 
C)  $\cos x$ 
D)  $- \cos x$ 
E)  1

Question 20

Find the exact value of the given expression using a sum or difference formula.
$\cos \frac { 7 \pi } { 12 }$

Accepted Answer

A)  $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
B)  $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
C)  $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
D)  $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
A)  $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
B)  $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$ 
C)  $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$ 
D)  $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$

Simplify the given expression algebraically. $\cos \left( \frac { \pi } { 2 } + x \right)$

Solve the multi-angle equation below. $\cos \left( \frac { x } { 2 } \right) = - \frac { \sqrt { 3 } } { 2 }$

Use the half-angle formulas to determine the exact value of the following. $\cos \left( - 22.5 ^ { \circ } \right)$

Verify the identity shown below. $\frac { 1 - \sin \theta } { 1 + \sin \theta } = 2 \sec ^ { 2 } \theta - 2 \sec \theta \tan \theta - 1$

If $x = 2 \cot \theta$ , use trigonometric substitution to write $\sqrt { 4 + x ^ { 2 } }$ as a trigonometric function of $\theta$ , where $0 < \theta < \pi$ .

Verify the identity shown below. $\frac { \tan \theta + 1 } { \sec \theta + \csc \theta } = \sin \theta$

Find all solutions of the following equation in the interval $[ 0,2 \pi )$ . $2 \cos x - \sec x = 0$

Verify the identity shown below. $\csc \theta - \cos \theta \cot \theta = \sin \theta$

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. $\cot \beta \sec \beta$

Use the sum-to-product formulas to write the given expression as a product. $\sin 9 \theta - \sin 7 \theta$

Find the exact value of the given expression using a sum or difference formula. $\sin 345 ^ { \circ }$

Find the exact value of the given expression using a sum or difference formula. $\sin 165 ^ { \circ }$

Determine which of the following are trigonometric identities. I. $\csc ( \theta ) \sec ( \theta ) = \tan ( \theta )$ II. $\csc ( \theta ) \tan ( \theta ) = \sec ( \theta )$ III. $\tan ( \theta ) \sec ( \theta ) = \csc ( \theta )$ IV. $\csc ( \theta ) \sin ( \theta ) = 1$

Write the given expression as algebraic expression. $\cos ( \arccos x - \arcsin x )$

Find the exact value of $\cos ( u - v )$ given that $\sin u = - \frac { 9 } { 41 }$ and $\cos v = \frac { 15 } { 17 }$ . (Both $u$ and $v$ are in Quadrant IV.)

Write the given expression as the sine of an angle. $\sin 85 ^ { \circ } \cos 50 ^ { \circ } + \sin 50 ^ { \circ } \cos 85 ^ { \circ }$

Verify the given identity. $\frac { \cos u - \cos v } { \cos u + \cos v } = - \tan \frac { 1 } { 2 } ( u + v ) \tan \frac { 1 } { 2 } ( u - v )$

Verify the given identity. $\cos ( x + y ) \cos ( x - y ) = \cos ^ { 2 } x - \sin ^ { 2 } y$

Simplify the given expression algebraically. $\cos ( \pi + x )$

Find the exact value of the given expression using a sum or difference formula. $\cos \frac { 7 \pi } { 12 }$

Functions and Their Graphs

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Exam 5: Analytic Trigonometry

Simplify the given expression algebraically. cos⁡(π2+x)\cos \left( \frac { \pi } { 2 } + x \right)cos(2π​+x)

Solve the multi-angle equation below. cos⁡(x2)=−32\cos \left( \frac { x } { 2 } \right) = - \frac { \sqrt { 3 } } { 2 }cos(2x​)=−23​​

Use the half-angle formulas to determine the exact value of the following. cos⁡(−22.5∘)\cos \left( - 22.5 ^ { \circ } \right)cos(−22.5∘)

Verify the identity shown below. 1−sin⁡θ1+sin⁡θ=2sec⁡2θ−2sec⁡θtan⁡θ−1\frac { 1 - \sin \theta } { 1 + \sin \theta } = 2 \sec ^ { 2 } \theta - 2 \sec \theta \tan \theta - 11+sinθ1−sinθ​=2sec2θ−2secθtanθ−1

If x=2cot⁡θx = 2 \cot \thetax=2cotθ , use trigonometric substitution to write 4+x2\sqrt { 4 + x ^ { 2 } }4+x2​ as a trigonometric function of θ\thetaθ , where 0<θ<π0 < \theta < \pi0<θ<π .

Verify the identity shown below. tan⁡θ+1sec⁡θ+csc⁡θ=sin⁡θ\frac { \tan \theta + 1 } { \sec \theta + \csc \theta } = \sin \thetasecθ+cscθtanθ+1​=sinθ

Find all solutions of the following equation in the interval [0,2π)[ 0,2 \pi )[0,2π) . 2cos⁡x−sec⁡x=02 \cos x - \sec x = 02cosx−secx=0

Verify the identity shown below. csc⁡θ−cos⁡θcot⁡θ=sin⁡θ\csc \theta - \cos \theta \cot \theta = \sin \thetacscθ−cosθcotθ=sinθ

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. cot⁡βsec⁡β\cot \beta \sec \betacotβsecβ

Use the sum-to-product formulas to write the given expression as a product. sin⁡9θ−sin⁡7θ\sin 9 \theta - \sin 7 \thetasin9θ−sin7θ

Find the exact value of the given expression using a sum or difference formula. sin⁡345∘\sin 345 ^ { \circ }sin345∘

Find the exact value of the given expression using a sum or difference formula. sin⁡165∘\sin 165 ^ { \circ }sin165∘

Write the given expression as algebraic expression. cos⁡(arccos⁡x−arcsin⁡x)\cos ( \arccos x - \arcsin x )cos(arccosx−arcsinx)

Find the exact value of cos⁡(u−v)\cos ( u - v )cos(u−v) given that sin⁡u=−941\sin u = - \frac { 9 } { 41 }sinu=−419​ and cos⁡v=1517\cos v = \frac { 15 } { 17 }cosv=1715​ . (Both uuu and vvv are in Quadrant IV.)

Write the given expression as the sine of an angle. sin⁡85∘cos⁡50∘+sin⁡50∘cos⁡85∘\sin 85 ^ { \circ } \cos 50 ^ { \circ } + \sin 50 ^ { \circ } \cos 85 ^ { \circ }sin85∘cos50∘+sin50∘cos85∘

Verify the given identity. cos⁡u−cos⁡vcos⁡u+cos⁡v=−tan⁡12(u+v)tan⁡12(u−v)\frac { \cos u - \cos v } { \cos u + \cos v } = - \tan \frac { 1 } { 2 } ( u + v ) \tan \frac { 1 } { 2 } ( u - v )cosu+cosvcosu−cosv​=−tan21​(u+v)tan21​(u−v)

Verify the given identity. cos⁡(x+y)cos⁡(x−y)=cos⁡2x−sin⁡2y\cos ( x + y ) \cos ( x - y ) = \cos ^ { 2 } x - \sin ^ { 2 } ycos(x+y)cos(x−y)=cos2x−sin2y

Simplify the given expression algebraically. cos⁡(π+x)\cos ( \pi + x )cos(π+x)

Find the exact value of the given expression using a sum or difference formula. cos⁡7π12\cos \frac { 7 \pi } { 12 }cos127π​

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Filters

Simplify the given expression algebraically. $\cos \left( \frac { \pi } { 2 } + x \right)$

Solve the multi-angle equation below. $\cos \left( \frac { x } { 2 } \right) = - \frac { \sqrt { 3 } } { 2 }$

Use the half-angle formulas to determine the exact value of the following. $\cos \left( - 22.5 ^ { \circ } \right)$

Verify the identity shown below. $\frac { 1 - \sin \theta } { 1 + \sin \theta } = 2 \sec ^ { 2 } \theta - 2 \sec \theta \tan \theta - 1$

If $x = 2 \cot \theta$ , use trigonometric substitution to write $\sqrt { 4 + x ^ { 2 } }$ as a trigonometric function of $\theta$ , where $0 < \theta < \pi$ .

Verify the identity shown below. $\frac { \tan \theta + 1 } { \sec \theta + \csc \theta } = \sin \theta$

Find all solutions of the following equation in the interval $[ 0,2 \pi )$ . $2 \cos x - \sec x = 0$

Verify the identity shown below. $\csc \theta - \cos \theta \cot \theta = \sin \theta$

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. $\cot \beta \sec \beta$

Use the sum-to-product formulas to write the given expression as a product. $\sin 9 \theta - \sin 7 \theta$

Find the exact value of the given expression using a sum or difference formula. $\sin 345 ^ { \circ }$

Find the exact value of the given expression using a sum or difference formula. $\sin 165 ^ { \circ }$

Write the given expression as algebraic expression. $\cos ( \arccos x - \arcsin x )$

Find the exact value of $\cos ( u - v )$ given that $\sin u = - \frac { 9 } { 41 }$ and $\cos v = \frac { 15 } { 17 }$ . (Both $u$ and $v$ are in Quadrant IV.)

Write the given expression as the sine of an angle. $\sin 85 ^ { \circ } \cos 50 ^ { \circ } + \sin 50 ^ { \circ } \cos 85 ^ { \circ }$

Verify the given identity. $\frac { \cos u - \cos v } { \cos u + \cos v } = - \tan \frac { 1 } { 2 } ( u + v ) \tan \frac { 1 } { 2 } ( u - v )$

Verify the given identity. $\cos ( x + y ) \cos ( x - y ) = \cos ^ { 2 } x - \sin ^ { 2 } y$

Simplify the given expression algebraically. $\cos ( \pi + x )$

Find the exact value of the given expression using a sum or difference formula. $\cos \frac { 7 \pi } { 12 }$