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.

  • SOH stands for Sine equals Opposite over Hypotenuse
  • CAH stands for Cosine equals Adjacent over Hypotenuse
  • TOA stands for Tangent equals Opposite over Adjacent
  • Graphs of Sine, Cosine, Tangent

    NOTE:

  • that the graph of tan has asymptotes (lines which the graph gets close to, but never crosses). These are the red lines (they aren't actually part of the graph);
  • that the graphs of sin, cos and tan are periodic. This means that they repeat themselves;
  • the symmetry of the graphs. For example, cos is symmetrical in the y-axis,
  • Graphs of Sine, Cosine, Tangent

    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





    Right-angle isosceles triangle: AB=CB Pythagoras theorem:
    (AB)2 + (CB)2 = (AC)2
    2(AB)2 = (AC)2

    Sine, Cosine, Tangent of special angles





    Equilateral triangle: AB=CB=AC=2
    BD=DC=1

    Pythagoras theorem: (AD)2 + (BD)2 = (BA)2
    (AD)2 + 12 = 22
    (AD)2 = 4-1=3

    Sine, Cosine, Tangent



    Sine, Cosine, Tangent



    Sine, Cosine, Tangent 1800





    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 α0 = 450 or in radians α0 = 0.7854…

    which is an approximation toα0 = π/4.


    However, from the graph we can find other solutions: …-3150, -1350, 450, 2250, 4050, 5850,…

    α0+n*1800; n=0;±1; ±2; ±3…



    Or in radians:





    Solution of Trigonometric Equations: cos(α)=p; -1≤p≤1

    Let us solve equation cos(α) = 0.5; One answer is α0 = 600 or in radians α0 = π/3

    The second solution α0= -600 or in radians α0 = -π/3

    However, from the graph we can find other solutions:
    ±α0+n*3600; n=0;±1; ±2; ±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 α0 = 300 or in radians α0 = π/6

    The second solution α0= 1800 - 300 = 1500
    or in radians α0 = 5π/6

    From the graph we can find other solutions:
    α0+n*3600; n=0;±1; ±2; ±3…



    18000+n*3600; 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,
    which makes them easy to remember.
    ;



    Trigonometric relations



    Dividing the Pythagorean identity through by either cos2(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 . Therefore, we will put 2α=θ, so that
    α becomes θ/2



    Sine, Cosine, Tangent





    Inverse Trigonometric Functions



    Function that is inverse to sin is called arcsine.
    The function arcsine is multi-value function. Example:





    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:
    Rational functions
    of sin and cos:

    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:
    Rational functions
    of sin and cos:

    R(sin(x), cos(x))=const;

  • If equation invariant with respect to substitution x→ π - x it easier to try to reduce to
    Rational function with respect to sin(x)
  • If equation invariant with respect to substitution x→ - x it easier to try to reduce to Rational function with respect to cos(x)
  • If equation invariant with respect to substitution x→ π + x it easier to try to reduce to
    Rational function with respect to tan(x)
  • Solution of trigonometric equations Some hints:
    Rational functions
    of sin and cos:



    Solution of trigonometric equations Some hints:
    Rational functions
    of sin and cos:

    R(sin(x)-cos(x), sin(x)cos(x))

    R(sin(x)+cos(x), sin(x)cos(x))



    Solution of trigonometric equations Some hints:
    Rational functions
    of sin and cos:

    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 t2. 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
    There are four angles in the range 0-2π: 0, π/2, π, 3π/2


  • For 0 : cos(0) + sin(0) = 1 + 0 = 1. This root coresponds to t=1
  • For π/2 : cos(π/2) + sin(π/2) = 0 + 1 = 1. This root coresponds to t=1
  • For π : cos(π) + sin(π) = -1 + 0 = -1. This root coresponds to t=-1
  • For 3π/2 : cos(3π/2) + sin(3π/2) = 0 - 1 = -1. This root coresponds to t=-1
  • Oscillations

    Let us suppose that a wheel of radius R is rotating anticlockwise with angular velocity ω radians per seconds.
    Let us assume that at time moment t0
    X-coordinate of a point B equals to R and its Y-coordinate equals to 0.
    As the angular velocity is ω radians per seconds angle AOB at time moment t will be equal ωt.
    So the coordinates of the point B will be determined by the following relations:





    We know that the period of cos and sin is . 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 . 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
    The number of complete rotations per second:

    Obviously

    Oscillations

    Let us consider the motion of one of projections of point B.
    Let us choose y-axis projection. Corresponding equation is:




    Such a periodic motion is an example of oscillatory motion or oscillations.
    The argument of sin(ωt) in any particular moment of time is called
    phase of oscillations. R is the amplitude and ω is an angular frequency of oscillations.
    Let us consider two different oscillations described by the following equations:

    Let us calculate phase difference between these two motions.
    We can rewrite:


    Therefore, the phase difference is

    radians.

    One motion repeats the other one after

    radians of phase or


    time units (seconds if is ω measured in radians per second).

    Oscillations and waves

    Oscillations

    Solution of trigonometric equations



    Solution of trigonometric equations