Finding The Area Of A Triangle Using Sine - Varsity Tutors

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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 approximationRemember 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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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).Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

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

A$6$B$12$C$\frac{3}{2}$D$\frac{1}{2}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

You are designing a triangular garden with sides $a = 5$, $b = 7$, and an included angle $C = 60^\circ$. Find the area.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

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

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

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

A$54$B$\frac{54\sqrt{3}}{4}$C$27\sqrt{3}$D$\frac{108}{\sqrt{3}}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

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