Law of Sines and Law of Cosines


 

Law of Sines



Section Overview

●    Understand and apply the Law of Sines to find unknown measurements in right and non-right triangles.

 

 

Key Vocabulary:

Degree - An angle measure.


Radian - Arc measurement.

 

The law of sines relates the ratios of side lengths of triangles to their respective opposite angles. This ratio remains equal for all three sides and opposite angles. We can therefore apply the sine rule to find the missing angle or side of any triangle using the requisite known data.

 

Law of Sines Formula:



Example 1: Given a = 20 units c = 25 units and Angle C = 42°. Find the angle A of the triangle.

Solution: For the given data, we can use the following formula of sine law: a/sinA = b/sinB = c/sinC

20/sin A = 25/sin 42°
sin A/20 = sin 42°/25
sin A = (sin 42º/25) × 20
sin A = (sin 42º/25) × 20
sin A = (0.6691/5) × 4
sin A = 0.5353
A = sin-1(0.5363)
A = 32.36°


Answer: ∠A = 32.36°

 

 

 

 

Example 2:  Solve for x.

 

 



Solution:

 


7 Sin(46.5°) = x sin (39.4°)
7 (0.725) = x (0.635)
5.078 = x (0.635)
x = 8

 

 

 

 

 

 

 

 


 

 

Let’s Practice.

Solve triangles using the law of sines (practice) | Khan Academy

 
Law of Cosines

 

 

 

 

 

Section Overview

·      Recall the law of cosines, a2 = b2 + c2 – 2bc cos A.

·      Recognize how the law of cosines can be applied to a triangle to determine an unknown side length (given a diagram or otherwise).

·      Recognize how the law of cosines can be applied to a triangle to determine an unknown angle measure (given a diagram or otherwise).

 

Key Vocabulary:

Initial side - Starting point.

Terminal side - Ending point.

Central Angle - Angle formed by 2 radii.

 

The Law of Cosines is used to find the remaining parts of an oblique (non-right) triangle when either the lengths of two sides and the measure of the included angle is known (SAS) or the lengths of the three sides (SSS) are known. In either of these cases, it is impossible to use the Law of Signs because we cannot set up a solvable proportion.


The Law of Cosine states that: c2 = a2 + b2 – 2ab cosC.


This resembles the Pythagorean Theorem except for the third term and if C is a right angle the third term equals 00 because the cosine of 90° is 0 and we get the Pythagorean Theorem.  So, the Pythagorean Theorem is a special case of the Law of Cosines.


The Law of Cosines can also be stated as: 

b2 = a2 + c2 – 2ac cosB     OR     a2 = b2 + c2 – 2bc cosA

 

 

Law of Cosines Formula:

 

 

 

 

 


Example 1: Two Sides and the Included Angle-SAS

 

Given a = 11, b = 5 and m∠C = 20°. Find the remaining side and angles.


 

 

c2 = a2 + b2 – 2ab cosC

 

   

      » 6.53

 

 



Example 2: Three Sides-SSS

 

Given a = 8, b = 19, and c = 14.  Find the measures of the angles.

 

 

It is best to find the angle opposite the longest side first.  In this case, that is side b.

 

 

 

 

Since cos B is negative, we know that B is an obtuse angle.

 

B » 116.80°

 

Since B is an obtuse angle and a triangle has at most one obtuse angle, we know that angle A and angle C are both acute.

  
To find the other two angles, it is simplest to use the Law of Sines. 

 



 

 

 

 

 

 

 

 

 


 

 

Let’s Practice.

Solve triangles using the law of cosines (practice) | Khan Academy