## Tan Graph – y=tan(x)

The graph of y=tanx is different from the other cos and sin graphs as it has a range from -∞ to ∞ and a period of 180° or π radians. The graph of y=tan(x) in radians is shown below

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.

## Auxiliary Angle Method for Solving Trigonometry Equations

This is a method of solving equations in the form asinx+bcosx = c where a and b are constants and c is another expression.

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

2^{2}+3^{2} = r^{2}

since sin^{2}+cos^{2} = 1

so r =√13

So we can write

2sinx + 3 cosx = √13 cos(x+56.3) = 3

so x = cos<sup>-1</sup>(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 = √(a^{2}+b^{2}) sin(x+tan^{-1}(b/a))

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

By David Woodford

## Compound Angles – sin(A+B) = cosAsinB+sinAcosB

Compound angles are angles made by adding two other angles together. When using trigonometry unfortunately you cant just “times out” the trig function but have to use an identity. This post will consider how we get the identity for sin(A+B):

**sin(A+B) = sinAcosB+sinBcosA**

From the definition of sin=opp/hyp we find

sin(A+B) = RT/OR

But sinceRT comprises of RS+ST

By David Woodford

## Sec, Cosec, Cot

Sec, cosec and cot are all functions in trigonometry. They are simply equal to one over on of the other functions, ie cos, sin and tan.

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)

se**c **goes with **c**os

co**s**ec goes with **s**in

co**t** goes with **t**an

## Trigonometry Identities

There a number of “identities” in trigonometry that can be found from the basic ideas of sin, cos and tan as explained in my earlier post. These identities can help in solving equations involving trig functions, especially when there are 2 or more different functions as the often allow you to write the equation in terms of one function, eg sin, that you can then solve. One of the identities is: **sin ^{2} + cos^{2} = 1. ** To prove this consider a right angled triangle with side a,b and c as shown below

From this we can use Pythagoras theorem to say: a^{2}+b^{2}=c^{2 now we know sin t = b/c so b = csin t cos t = a/c so a = ccos t} substituting these values in the above equation we get (csint)^{2} +(ccost)^{2} = c^{2} canceling the c^{2} we get **sint ^{2} + cost^{2} = 1**

There are trig functions that are equal to 1 over sin, cos and tan called cosec = 1/sin, sec = 1/cos and cot = 1/tan. These can be remembered using the third letter rule as the third letter of each of these corresponds to the the function it is one over.

Using these a cos

^{2}+ sin

^{2}= 1 we can calculate other identities

**tan**We can obtain this by dividing through by cos

^{2}t + 1 = sec^{2}t^{2}as we know sin/cos = tan, cos/cos = 1 and 1/cos = sec. Other similar identities can be obtained for cosec and cot.

## Hyperbolic Functions

**This post can new be viewed from www.breakingwave.co.nr :):)**

Hyperbolic functions are similar to sin,cos tan etc in trigonometry and share many similar rules. Usually hyperbolic functions are written like the trigonometric ones but with a h on the end, eg sinh and cosh.

The hyperbolic functions can be all written in terms of e, sinh and cosh are as follows

sinh(x) = (e^{x} – e^{-x})/2

cosh(x) = (e^{x} + e^{-x})/2

And tanh can be defined as sinh/cosh so

tanh = (e^{x} – e^{-x}) / (e^{x} + e^{-x})

though this is often written as

tanh = (e^{2x} – 1) / (e^{2x} + 1)

by timesing the top and bottom by e^{x}

the other other hyperbolic functions sinh as sech, coth etc can be found in the same way as they would be in trigonometry, by using 1 over the other functions, ie sech = 1/cosh

Most of the identities in trigonometry have a similar identity with hyperbolic functions, however in most of these whenever there is a sin^{2} it changes to a -sinh^{2}

so

cosh^{2} – sinh^{2}=1

which you can work out by placing the equations with e’s in the place of sinh and cosh

IF you have any questions pleas ask below

## Tan = sin/cos

this site is now at www.breakingwave.co.nr

This is often useful when solving trig equations so i thought i’d include it

basically:

sin = opp/hyp

and

cos=adj/hyp

so

sin/cos = ^{(opp/hyp)}/_{(adj/hyp)}

so if we cancel the hyp’s we get

sin/cos = opp/adj

and since tan = opp/adj

**tan = sin/cos**

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