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Finding the Area of a Triangle using Sine

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Finding the Area of a Triangle using Sine

Study Guide

Key Definition

Let R represent the area of the triangle. The area can be found using the formula $R = \frac{1}{2}bc\sin(A)$, where $b$ and $c$ are side lengths and $A$ is the included angle.

Important Notes

Using sine helps find the area when height is not directly known.
The formula $R = \frac{1}{2}bc\sin(A)$ is useful for non-right triangles.
$\sin(A)$ is the sine of angle $A$.
The formula can be adapted to use other angles by adjusting the sides accordingly.
The formula works with angles in degrees or radians; ensure the calculator mode matches the angle unit.

Mathematical Notation

$\frac{a}{b}$ represents division

$+$ represents addition

$-$ represents subtraction

$\times$ represents multiplication

$\sin(\theta)$ is the sine function

$√$ represents the square root

$°$ denotes degrees

$≈$ denotes approximation

Remember to use proper notation when solving problems

Why It Works

Using the sine function allows us to calculate the area of a triangle by relating the sides and the angle between them, even when the height is not known. This utilizes the property $h = c\sin(A)$.

Remember

Always use the formula $R = \frac{1}{2}bc\sin(A)$ for triangles where the height is not known directly.

Quick Reference

Area Formula:$R = \frac{1}{2}bc\sin(A)$

Understanding Finding the Area of a Triangle using Sine

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Video explanation of this concept

Beginner

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Beginner Explanation

We recall that the area R of a triangle is given by R = \frac{1}{2}bh, and we use h = c\sin(A) to derive the formula R = \frac{1}{2}bc\sin(A).

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the area of a triangle with sides $b = 3$, $c = 4$ and angle $A = 90^\circ$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You are designing a triangular garden with sides $a = 5$, $b = 7$, and an included angle $C = 60^\circ$. Find the area.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Calculate the area of a triangle with sides $a = 8$, $c = 10$, and angle $B = 45^\circ$.

Challenge Quiz

Single Choice Quiz

Advanced

For a triangle with sides $a = 9$, $b = 12$, and angle $C = 120^\circ$, find the area.

Recap

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Review key concepts and takeaways