# Thread: a couple of questions...thanks

1. ## a couple of questions...thanks

I dont know how to find the exact value of something(trig)...e.g.

sec(13Pi/6) I get a long decimal(1.31...) but how am I meant to get the exact value answer which in this case is 2/3 of the square root of 3. (which is 1.31 when u calculate it but when I see 1.31 how am I meant to know that its 2/3 the square root of 3?

question 2:
3(tan^2 )theta - sec theta = 1

for 0 <= theta =< 2PI

can anyone solve that for me? thanks

oh and for the following graph sketches:

y = sin3x
y = |sin3x|
y = sin|3x|

so like I was confused to whether they amplitude was 3 times bigger or the frequency is like 3 times smaller for them. can someone tell me what the difference of those curves are from a normal sinx

2. Originally Posted by grammar
I dont know how to find the exact value of something(trig)...e.g.

sec(13Pi/6) I get a long decimal(1.31...) but how am I meant to get the exact value answer which in this case is 2/3 of the square root of 3. (which is 1.31 when u calculate it but when I see 1.31 how am I meant to know that its 2/3 the square root of 3?

[snip]
$\sec \left(\frac{13 \pi}{6} \right) = \sec \left( \frac{(12 + 1) \pi}{6} \right) = \sec \left(\frac{12 \pi}{6} + \frac{\pi}{6}\right)$

$= \sec \left(2 \pi + \frac{\pi}{6}\right) = \sec \left(\frac{\pi}{6}\right) = \frac{1}{\cos \left(\frac{\pi}{6}\right)} = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}}$.

3. Originally Posted by grammar
[snip]
question 2:
3(tan^2 )theta - sec theta = 1

for 0 <= theta =< 2PI

can anyone solve that for me? thanks

[snip]
Substitute (from the Pythagorean Identity) $\tan^2 \theta = \sec^2 \theta - 1$:

$3 (\sec^2 \theta - 1) - \sec \theta = 1 \Rightarrow 3 \sec^2 \theta - \sec \theta - 4 = 0$.

Let $x = \sec \theta$ to see how to factorise:

$3x^2 - x - 4 = 0 \Rightarrow (3x - 4)(x + 1) = 0$.

Therefore either $x = \frac{4}{3}$ or $x = -1$.

Therefore either $\sec \theta = \frac{4}{3}$ or $\sec \theta = -1$.

The latter can be exactly solved easily. The former can only be found as either a generic exact answer or a decimal approximation.

4. Originally Posted by grammar
[snip]
oh and for the following graph sketches:

y = sin3x
y = |sin3x|
y = sin|3x|

so like I was confused to whether they amplitude was 3 times bigger or the frequency is like 3 times smaller for them. can someone tell me what the difference of those curves are from a normal sinx
Draw the graph of $y = |\sin (3x)|$ by reflecting around the x-axis the parts of the graph of $y = \sin (3x)$ that lie below the x-axis. So there are salient points at $(0, \, \pm n \pi)$ where n is an integer.

Draw the graph of $y = \sin |3x|$ by reflecting around the x-axis the part of the graph of $y = \sin (3x)$ that lies to the left of the y-axis. So there's a salient point at (0, 0).

Note: $\sin |3x| = \sin (3x)$ for $x \geq 0$ and $\sin |3x| = \sin (-3x) = -\sin (3x)$ for $x < 0$.

Salient point: Salient Point -- from Wolfram MathWorld