Double-Angle Formulas

l flipboi l · #1 $Old$ November 2nd, 2009, 02:49 PM

Hello,

Can someone please help me with these problems?

Find Sin2x, Cos 2x, and Tan2x for:

1. tan x= -4/3 quadrant II
2. csc x = 4, tan x<0

how do I start the problem?

for 1, is Sin -4/1 and Cos 3/1
and for 2 is sin 1/4?

Thanks

Grandad · #2 $Old$ November 3rd, 2009, 08:01 AM

Hello l flipboi l

Quote:

Originally Posted by l flipboi l $View Post$

Hello,

Can someone please help me with these problems?

Find Sin2x, Cos 2x, and Tan2x for:

1. tan x= -4/3 quadrant II
2. csc x = 4, tan x<0

how do I start the problem?

for 1, is Sin -4/1 and Cos 3/1
and for 2 is sin 1/4?

Thanks

Are you familiar with this diagram that tells you when the various trig functions are positive?

$\begin{array}{c|c}S & A\\ \hline T & C \end{array}$

(Remember ACTS starting in QI and going clockwise.)

1. In QII, only sine is positive. So if $\tan x = -\frac43$ and $x$ is in QII, $\sin x = \frac45$ and $\cos x = \frac35$ (using Pythagoras). You should be able to find the sine, cosine and tangent of $2x$ using the standard double-angle formulae.

2. You are right: $\csc x = \frac{1}{\sin x}= 4 \Rightarrow \sin x = \frac14$ . Since sine is positive, we are either in QI or QII. So the fact that $\tan x < 0$ means that $x$ must be in QII. Using Pythagoras, then:

$\tan x = -\frac{1}{\sqrt{15}}$ and $\cos x = -\frac{\sqrt{15}}{4}$

Again, the standard double-angle formula will give you the results you need.

Grandad

l flipboi l · #3 $Old$ November 3rd, 2009, 11:15 AM

Quote:

Originally Posted by Grandad $View Post$

Hello l flipboi l
Are you familiar with this diagram that tells you when the various trig functions are positive?

$\begin{array}{c|c}S & A\\ \hline T & C \end{array}$

(Remember ACTS starting in QI and going clockwise.)

1. In QII, only sine is positive. So if $\tan x = -\frac43$ and $x$ is in QII, $\sin x = \frac45$ and $\cos x = \frac35$ (using Pythagoras). You should be able to find the sine, cosine and tangent of $2x$ using the standard double-angle formulae.

2. You are right: $\csc x = \frac{1}{\sin x}= 4 \Rightarrow \sin x = \frac14$ . Since sine is positive, we are either in QI or QII. So the fact that $\tan x < 0$ means that $x$ must be in QII. Using Pythagoras, then:

$\tan x = -\frac{1}{\sqrt{15}}$ and $\cos x = -\frac{\sqrt{15}}{4}$

Again, the standard double-angle formula will give you the results you need.

Grandad

Thanks Grandad!