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Essential Idea :Exploring the Sine and Cosine Rules: Investigating the
DP Math: AA SL 11
Role of Mathematical Principles in Analyzing Geometric Relationships and
ATL:Communication
Solving Real-World Problems
Example
In the triangle
ABC,AB=6 and angle
BAC=(pi)/(3),
BD is the arc of a circle, centre
A, and
BC is a tangent to the circle.
Find the area of the shaded region
BCD.
Solution:

Essential Idea :Exploring the Sine and Cosine Rules: Investigating the $\newline$ DP Math: AA SL $11$ $\newline$ Role of Mathematical Principles in Analyzing Geometric Relationships and $\newline$ ATL:Communication $\newline$ Solving Real-World Problems $\newline$ Example $\newline$ In the triangle $A B C, A B=6$ and angle $B A C=\frac{\pi}{3}$ , $B D$ is the arc of a circle, centre $A$ , and $B C$ is a tangent to the circle. $\newline$ Find the area of the shaded region $B C D$ . $\newline$ Solution:

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Write equations of cosine functions using properties

Full solution

Q. Essential Idea :Exploring the Sine and Cosine Rules: Investigating the

\newline

DP Math: AA SL

11

\newline

Role of Mathematical Principles in Analyzing Geometric Relationships and

\newline

ATL:Communication

\newline

Solving Real-World Problems

\newline

Example

\newline

In the triangle

A B C, A B=6

and angle

B A C=\frac{\pi}{3}

,

B D

is the arc of a circle, centre

A

, and

B C

is a tangent to the circle.

\newline

Find the area of the shaded region

B C D

.

\newline

Solution:

Calculate Circle Radius: First, calculate the radius of the circle. Since BD is an arc with center A and angle $BAC$ is $\pi/3$ , the radius $r$ can be found using the formula for the length of an arc, $s = r\theta$ , where $\theta$ is in radians. Here, $AB$ is the radius, so $r = 6$ .
Find Length of Arc BD: Next, calculate the length of arc BD. The formula for the length of an arc is $s = r\theta$ . Substituting $r = 6$ and $\theta = \pi/3$ , we get $s = 6 \times (\pi/3) = 2\pi$ .

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