There are two solutions to sin(x)=0.65 we get solution (1) when we take the inverse but we need a way of finding (2) from (1)

The four sections of the CAST are cos,all,sin and tan starting by labelling the bottom right and working round in an anti-clockwise direction.

Use the following method to work out the values for the inverse between 0 and 2pi or 360

1. find the value of the inverse of the positive between 0 and 90 or pi/2 using a normal method (eg a calculator).
   So if we want to find sin^-1(0.6) we calculate 0.6435 rads or 36.87 deg and if we want sin^-1(-0.8) we calculate sin^-1(0.8) to get 53.13 deg or 0.927 rads
2. draw in the four lines on the CAST diagram (shown in green) that represent the angle. Do this by measuring from the horizontal the angle calculated in 1
3. If the value we are finding the inverse for is negative (eg we are finding sin^-1(-0.6)) consider the quadrants that don’t include the name of the function. (so if we are finding inverse sin only consider Tan and Cos, if we are finding inverse cos only consider sin and tan and if we are finding inverse tan only consider sin and cos)
4. If the value we are finding the inverse for is positive (eg we are finding sin^-1(0.6)) only consider the quadrants with the name of the function and all. (so if we are finding inverse sin we only consider sin and all, if we are finding inverse cos only consider only consider cos and all and if we are finding inverse ta only consider tan and all)
5. Calculate the angle from the zero line anti-clockwise to the lines in the quadrant’s we are considering. The angle labels I’ve put on the axes should make this easier. These values are your solutions.

All four lines representing the angle x drawn but we will only be interested in two of them

You can check that this works by putting the values back into you calculator and if you want to check that you’ve got all of the solutions check against the graph of the function.

Example

Find all the solutions of cos(x) = 0.7 between 0 and 360^o or rads.

We can use a calculator to find the value between 0 and 90 deg or pi/2 rads
so write y = cos^-1(0.7) = 45.57 deg (we’ll work in degrees for the example to avoid repeating all the calculations)

We no draw this on the cast diagram and choose the two lines we need (in red), in this case the lines in the cos and “all” quadrants.

We then calculate the anti-clockwise angles to these lines from the horizontal:
so we get 45.57 deg for the in the “all” quadrant and 360-45.57=314.43 for the cos quadrant
so our set of solutions for x between 0 and 360 of cos(x)=0.7 are 45.57 and 314.43

Also see

• Sec, Cosec, Cot (0)
• Tan = sin/cos (6)
• Sin, Cos and Tan (15)
• Integrate cos and sin squared using double angles (2)
• The Chain Rule (0)
• Trigonometry Identities (11)

Using the identity tanx = sinx / cosx we can write

we can now substitue in
sin(A+B) = sinAcosB + sinBcosA
and
cos(A+B) = cosAcosB – sinAsinB
to get

We can now divide both the top and bottom by cosAcosB to get

or

We can now simplify this by cancelling any cosA and cosB to get

finally by substituting the identity we find our result

And it can be shown that this result can be extended to

Also see

• CAST Diagarams for finding values of sin, cos and tan between 0 and 360 or 2 pi (0)
• Tan Graph – y=tan(x) (0)
• Compound Angles – sin(A+B) = cosAsinB+sinAcosB (6)
• Sec, Cosec, Cot (0)
• Tan = sin/cos (6)
• Sin, Cos and Tan (15)

The Maclaurin series for a function f(x) is as follows:

…
or

Example sin(x)

An expression for sin(x) can be calculated using this method. Firstly we must decide what level of accuracy to go to, for the example we will go to the 3rdterm. The next step is to differentiate sinx. Firstly let f(x) = sinx

then

We can now combine these into the series to get

…

…

which can be used to calculate the value of sin(x) — though only for the radian measure of angle.

eg)
… 0.8663

Which is approximately the value you would get if you type sin(pi/3) into a calculator.

Also see

• Taylor Series with example cos(x) (1)

Graph of y=tan(x) in radians

As can be seen from the graph the curve passes through the origin. It has vertical asymtopes (lines it tends toward but never touches — in this case where the graph goes to infinity) at x =π/2,3π/2,5π/2 and x=-π/2,-3π/2 etc radians or at x=90,270,450 and x=-90,-270 etc degrees.

The graph has a stationary (flat) point whenver it crosses the x-axis.

Also see

• Sin, Cos and Tan (15)
• Sec, Cosec, Cot (0)
• Tan = sin/cos (6)
• CAST Diagarams for finding values of sin, cos and tan between 0 and 360 or 2 pi (0)
• Find equation of tangent to a curve (2)
• Online Graph Sketchers (0)

y = cos(x) - in radians

Unlike the sine graph the cosine graph is an even function as it is symmetrical about the y axis. It has a maximum value of 1 and a minimum value of -1

By David Woodford

Also see

• Sine and Cos Graphs Differentiating sin and cos (7)
• Trigonometry Identities (11)
• CAST Diagarams for finding values of sin, cos and tan between 0 and 360 or 2 pi (0)
• Integrate cos and sin squared using double angles (2)
• The Chain Rule (0)
• Sec, Cosec, Cot (0)

Compound angle of A+B showing how they relate

From the definition of cos we find

cos(A+B) = OT/OR

but
OT = OP – PT
and PT = SQ so
OT = OP – SQ

so

cos(A+B) = ( OP – SQ ) / OR
so
cos(A+B) = OP/OR – SQ/OR

if we now times the both the top and bottom of the first term by OQ and do the same for the second term but with RQ we can get

but OP/OQ = cosB,
OQ/OR = cosA,
SQ/RQ = sinB,
RQ/OR = sinA

so we get, when these are substituted in and re arranged

cos(A+B) = cosAcosB-sinAsinB

Also see

• Trigonometry Identities (11)
• Proof of Cosine Rule (4)
• Product Rule for Differentiation (0)
• Why the proof 2=1 is wrong (0)
• Tan Graph – y=tan(x) (0)
• Differentiate Logs with Proof (0)

On the graph below the angle, in radians, is along the x axis and the value of the sine function for that angle is on the y axis.

Graph of y=sin(x)

Also see

• Sine and Cos Graphs Differentiating sin and cos (7)
• Differentiate Inverse Cos – Proof (8)
• Trigonometry Identities (11)
• Sin, Cos and Tan (15)
• CAST Diagarams for finding values of sin, cos and tan between 0 and 360 or 2 pi (0)
• Integrate cos and sin squared using double angles (2)

It involves rewriting letting asinx + bcosx = rsin(x+y) (or you could use cos(x+y)) where y is acute and then finding values for r and y, then with only one trig function to deal with the equation can be solved more easily.

For example

consider 2sinx + 3cosx = 3

Let 2sinx + 3cosx = rsin(x+y)

Now expand the sin(x+y) to get

2sinx + 3cosx = rsinx cosy + rcosx siny

Since y is constant and therefore cosy and sin y are constant we can compare the coefficients to get

2 = rcosy —–(1)
3 = rsiny ——(2)

We can solve these to find values for r and y.
To find y consider (2)/(1) to get

3/2 = tany
since sin/cos = tan and the r’s cancel
so y = 56.3 °

To find r consider (1)2+(2)2 to get
22+32 = r2
since sin2+cos2 = 1
so r =√13

So we can write

2sinx + 3 cosx = √13 cos(x+56.3) = 3

so x = cos^-1(3/√13) -56.3

so x = cos-1(3/√13) -56.3

since cos-1(3/√13) = 33.7 for solutions between 0° and 90°

x = ±33.7 -56.3 + 180n where n is an integer

In General

asinx + bcosx = √(a2+b2) sin(x+tan-1(b/a))

If you have any questions, comments or corrections please leave them as a comment below

By David Woodford

Also see

• Trigonometry Identities (11)
• Proof of Cosine Rule (4)
• Understand the Sine and Cosine Rules (11)
• Area of a Triangle (3)
• Proof by Contradiction (0)
• Rational and Irrational Numbers (5)

sin(A+B) = sinAcosB+sinBcosA

Compound angle of A+B showing how they relate

From the definition of sin=opp/hyp we find

sin(A+B) = RT/OR

But sinceRT comprises of RS+ST

By David Woodford

Also see

• Trigonometry Identities (11)
• Sin, Cos and Tan (15)
• Auxiliary Angle Method for Solving Trigonometry Equations (1)
• Sec, Cosec, Cot (0)
• Hyperbolic Functions (0)
• Tan = sin/cos (6)

so

Sec = 1/cos

Cosec = 1/sin

cot = 1/tan

You can remember which is paired with which using the third letter rule. This is that the third letter is the first letter of the corresponding function ie)

sec goes with cos
cosec goes with sin
cot goes with tan

Also see

• Tan = sin/cos (6)
• Sin, Cos and Tan (15)
• CAST Diagarams for finding values of sin, cos and tan between 0 and 360 or 2 pi (0)
• Trigonometry Identities (11)
• Integrate cos and sin squared using double angles (2)
• The Chain Rule (0)