Adding vectors using trigonometric methods

A) Pythagoras's Theorem & Trigonometry 

noteReview: Mathematical concepts in trigonometry

Recall Pythagoras's Theorem: C2 = A2  + B

This theorem applies only to right angle triangles like the one below.  
In other words, it can be used only when vector A is perpendicular to vector B

right angle triangle

The three basic trigonometric identities that can be derived from a right angle triangle whose included angle is theta, are:

trig functions

 B) Sine and Cosine Laws Method

 noteFor triangles that are NOT right-angle triangles we can use modified trigonometric laws known as the cosine and sine laws:

The triangle below is a generalized isosceles triangle where none of the sides are perpendicular to one another.

noteNote that we use lower case letters to represent the sides of the triangle and upper case letters to represent the included angles.

non right angle triangle

noteMemorize these two laws ... they will come in very handy

  1. The sine Law
    sine cosinelaws1
  2. The Cosine Law

    sine cosinelaws2

Example: What was the total displacement of a balloon that drifts for 7 km in a [NE] direction and then is pushed by a wind in a southerly direction for another 3 km.

noteIn case you forgot: [NE] is a direction half way between the y axis [N] and the x axis  [E] ---> [N450E]

noteUse the sine/cosine laws to solve this problem.


As usual we will start with a quick sketch

noteKnowing the rules of angles here is also important.  For example the Opposite Angles Theorem tells us that the angle between d1 and d2 (angle B) is also  450 - see diagram below:

sine cosinelaws example

noteStep-by-step ...

  1. We first apply  the cosine Law to find the magnitude of the resultant vector, dtotal


b = dtotal (to be calculated)

a = 3 km [S];

c = 7 km [NE]

 B = 450

therefore applying the cosine law

sine cosinelaws2

b2 = (3)2 + ( 7)2 - 2(3)(7)(cos(450))

and b2 = 28.3 

and [ by taking the square root of b2]

b = 5.32 km   ----> this is the magnitude of dtotal


2. Finding the direction of the resultant - Apply the Sine Law to find the value of the angletheta

from the general equation sine cosinelaws1

we can only use two of the terms for which we have information:

sine example1

cross multiply, isolate,  and solve for sintheta

sine example2

sin theta = 0.398

 Now use the inverse of sin (sin-1) to find the angle in degrees


= invsin(0.398) = 23.500

3. Make a final statement indicating your findings

 therefore symbol the vector  dtotal has direction (450 + 23.50) --- see diagram below

final answer

therefore symbol dtotal = 5.32 km [N68.50]