Trigonometry
Dr Viktor Fedun
v.fedun@sheffield.ac.uk
The first trigonometric table was apparently compiled by Hipparchus, who is now consequently known as "the father of trigonometry." 
He also introduced the division of a circle into 360 degrees into Greece." 
How problem of estimating the height of tall and distant objects has been solved 2600 years ago ? 
PQ represents a tower which casts a shadow QC on level ground. AB is a stick placed vertically into the ground so that points C, A and P lie on a straight line. Triangles PQC and ABC are similar and 
This constant ratio is a property of an angle and is called the Tangent of the angle α or tan for short 
Sine, Cosine, Tangent 
Divide the length of one side by another side ... but you must know which sides! 
Sine, Cosine, Tangent 

SOHCAHTOA A way of remembering how to compute the sine, cosine, and tangent of an angle. 
Graphs of Sine, Cosine, Tangent 
NOTE: 
Graphs of Sine, Cosine, Tangent 
Sine, Cosine, Tangent and the Unit Circle from the Wolfram Demonstrations Project by Eric Schulz 
Radians and degrees 
The concept of radian measure, as opposed to the degree of an angle, is credited to Roger Cotes in 1714 
A complete revolution is 2π radians Conversion between radians and grads 
A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle. The angle Θ in radians is defined as: 
Most Common Angles 
The most common angles that appear in trigonometry, engineering and physics problems. 
Sine, Cosine, Tangent of special angles 
Rightangle isosceles triangle: AB=CB
Pythagoras theorem:

Sine, Cosine, Tangent of special angles 
Equilateral triangle: AB=CB=AC=2

Sine, Cosine, Tangent 
Sine, Cosine, Tangent 
Sine, Cosine, Tangent 180^{0}α 
Sine, Cosine, Tangent of negative angles 
Sine, Cosine, Tangent 
Sine, Cosine, Tangent: CAST rule 
The ratios which are positive in each quadrant are given by the Rule known as CAST rule. Otherwise the ratio is negative in sign. 
Graphs of Sine 
Graphs of Cosine 
Graphs of Tangent 
Solution of Trigonometric Equations: tan(α)=p 
Let us solve equation tan(α) = 1;
One answer is α_{0} = 45^{0} or in radians
α_{0} = 0.7854…
Or in radians: 

Solution of Trigonometric Equations: cos(α)=p; 1≤p≤1 
Let us solve equation cos(α) = 0.5;
One answer is α_{0} = 60^{0} or in radians
α_{0} = π/3
Or in radians: 

Solution of Trigonometric Equations: sin(α)=p; 1≤p≤1 
Let us solve equation sin(α) = 0.5;
One answer is α_{0} = 30^{0} or in radians
α_{0} = π/6
180^{0}α_{0}+n*360^{0}; n=0;±1; ±2; ±3… Or in radians: 

A useful mnemonic for certain values of sines and cosines 
For certain simple angles, the sines and cosines take the form
√ n
/2 for 0 ≤ n ≤ 4,


Trigonometric relations 
Dividing the Pythagorean identity through by either cos^{2}(θ) yields other identity 
The sum and difference identities 


The sum and difference identities 


The sum and difference identities 


The sum and difference identities 



Trigonometric relations 


Trigonometric relations 

Now, α is half of
2α. Therefore, we will put 2α=θ, so that

Sine, Cosine, Tangent 


Inverse Trigonometric Functions 

Function that is inverse to sin is called
arcsine. 
Inverse Trigonometric Functions 

Function that is inverse to tan is called arctan. 
Inverse Trigonometric Functions 

Solution of trigonometric equations 


Solution of trigonometric equations 


Solution of trigonometric equations
Some hints: R(sin(x), cos(x))=const; 
Substitution 
Rational function: In the case of one variable, x, a function is called a Rational function if and only if it can be written in the form where P and Q are polynomial functions in x and Q is not the zero polynomial. 
Solution of trigonometric equations
Some hints: R(sin(x), cos(x))=const; 
Rational function with respect to sin(x) Rational function with respect to tan(x) 
Solution of trigonometric equations
Some hints: 

Solution of trigonometric equations
Some hints: R(sin(x)cos(x), sin(x)cos(x)) R(sin(x)+cos(x), sin(x)cos(x)) 
Solution of trigonometric equations
Some hints: R(sin(x)cos(x), sin(x)cos(x)) R(sin(x)+cos(x), sin(x)cos(x)) 
EXAMPLE 1a 
EXAMPLE 1b 
EXAMPLE 1c 

EXAMPLE 2 


EXAMPLE 3 


However the expression of sin(2x) depends upon t^{2}.
Therefore roots that corresponds both to t=1 and t=1 are included in
πn/2. We need to separate roots that correspond to t=1 
Oscillations 
Let us suppose that a wheel of radius R is
rotating anticlockwise with angular velocity ω radians per seconds. 

We know that the period of cos and sin is 2π. One complete revolution of the wheel will take the same time T as for an argument of cos and sin in above equations to change from 0 to 2π. In our example the argument is ωt. This time T can be determined from : 


T is called period of rotation; ω
is called angular frequency 

Obviously 
Oscillations 
Let us consider the motion of one of projections of point B. 

Such a periodic motion is an example of oscillatory motion or oscillations. 

Let us calculate phase difference between these two motions. Therefore, the phase difference is One motion repeats the other one after time units (seconds if is ω measured in radians per second). 
Oscillations and waves 
Oscillations 
Solution of trigonometric equations 

Solution of trigonometric equations 
