Recap of Trigonometry in Right Angled Triangles

The trigonometric ratios provide a relationship between angles and lengths:

Exact Values

We need to know the exact values for each of the trigonometric functions with angles π/4, π/3 and π/6.

We can draw an equilateral triangle and a right-angled isosceles triangle to work these out.

We can use a table to help us remember them.

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Angles and Quadrants

In order to calculate trigonometric values of angles greater than 90 degrees we must measure the angle in a standard way. We measure it anticlockwise from the line OX.  Depending on the size of the angle we say it is in Quadrant I, Quadrant II, Quadrant II or Quadrant IV.

The acute angle made with the x-axis of any angle is called the basic angle.

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Trigonometric ratios of any angle

We can extend our original definition of the trigonometric functions by rotating r around a unit circle so that cos𝜽 is simply the x-coordinate, sin𝜽 is the y-coordinate and tan𝜽 is the gradient.

It is useful to remember the sign (i.e. + or -) of each of the trigonometric ratios in each quadrant.  Then to calculate a trigonometric ratio, we can calculate the basic angle (angle from x) and then use the quadrant to determine the sign.

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Trigonometric Graphs

Because the range of a periodic function is periodic, it can also be represented by a graph, with the range plotted on the y-axis and the domain plotted on the x-axis.

• So the x-axis will show the angle and the y-axis will show:
  • Vertical displacement for sine;
  • Horizontal displacement for cosine; and
  • Gradient for tangent.

So what will each of the principal trigonometric graphs look like?  Can you see any relationships between them?

Sine and cosine both have a period of 2π and tangent has a period of π.

Amplitude, in mathematics, is the distance between the average and extreme point of the wave, so sine and cosine both have amplitude 1.

Cosine is an even function; it is symmetrical about the y-axis.  This means that cos(-x) = cos(x).

Sine is an odd function; it has rotational symmetry of π around the origin.  We can see that sin(-x) = -sin(x).

Because of the period (or thinking of the unit circle definition), it is clear that, for all x, sin(x+2π)=sin(x), cos(x+2π)=cos(x), tan(x+π)=tan(x).

• We can also see on the graphs (considering horizontal translation), that:
  • sin(x+π)=-sin(x) and cos(x+π)=-cos(x); 
  • sin(π-x) = -sin(x-π) = -sin(-x) = sin(x); and
  • cos(π-x) = cos (x-π) = -cos(x).

Transformations

• As with all other functions (previously studied in the function topic):
  • y=asinx is the graph of y=sinx vertically stretched by a from the x-axis, giving a graph with amplitude a and period 2π;
  • y=sin(ax) is the graph of y=sinx horizontally stretched by 1/a from the y-axis, giving a graph with amplitude 1 and period 2π/a;
  • y=a+sinx is the graph of y=sinx vertically translated by a, amplitude and period unchanged; and
  • y= sin(x+a) is the graph of y=sinx horizontally translated by a, amplitude and period unchanged.

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Inverse Trigonometric Functions

Because trigonometric functions are many-one, we need to restrict their domain to define inverse functions. (It is from this restricted range that a calculator will give the output for an inverse trigonometric function. This value is called the principal value.

Considering the graphs of each of the three functions, what do you think would be a sensible domain restriction to apply for each one?

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Solving Trigonometric Equations

We can solve trigonometric equations, but due to the periodic nature of trigonometric functions, our solution will be a set of infinitely many values.  Cambridge questions will ask for those values within a specific domain.

The easiest way to solve trigonometric equations is to memorise the following three formulae and then insert the necessary values of k for the required domain

• In addition to these, we should also not forget the following two basic trigonometric identities:
  • sin²x+cos²x=1 for all x;
  • for all x provided cos x ≠ 0.

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Trigonometric Identities

We prove an identity by writing down one side of the identity, and systematically applying legitimate algebraic operations to it until we finally arrive at the other side of the identity.

It is normally easiest to start with the “more complicated side of the identity.

Our working should look like this:

LHS = …

=…

= …

= …

= …

= RHS

• There are many, many trigonometric identities, but for now we will use only the following two, which can be manipulated to prove a given identity: 
  • sin²x+cos²x=1 for all x;
  • for all x provided cos x ≠ 0.

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Solving Harder Equations (using trigonometric identities)

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