Solving Trigonometric Functions

[Ohxsnap]

Please help. I cannot do my homework. I have no examples to work from and my teacher did not help me with it because "she did not have time." Will someone please help me wth some examples. Please..

Here's a few problems...

SOLVE FOR 0 ? ? < 2?
We're working on solving trigonometric functions.(Also this is using the trigonometric identities and formulas) Answers will be coordinates of a radian on the Unit Circle.
Here is the unit circle..
http://www.math-tutoring-connection.com/images/unit_circle.gif
The coordinates are listed here.. also on the unit circle.


Here's some problems..

1. sin?cos? - cos? = 0

2. sec?csc?= csc?

3. cos²? + cos? - 6 = 0

Please help.. you have no idea what it means to me for your help. I greatly appreciate any ideas..

 

[wjm11]

1. sin?cos? - cos? = 0

2. sec?csc?= csc?

3. cos²? + cos? - 6 = 0

sin?cos? - cos? = 0
Factor:
cos?(sin? –1) = 0
So
cos? = 0
and
(sin? –1) = 0
sin? = 1

You can solve those, right?

sec?csc?= csc?
Divide both sides by csc?:

sec? = 1
1/sec? = 1
cos? = 1

You can solve that, too, right?

cos²? + cos? - 6 = 0

This is just a quadratic equation. Let’s factor it:

(cos? – 2)(cos? + 3) = 0
so
(cos? – 2) = 0
(cos? + 3) = 0
Rearranging, we get
cos? = 2
cos? = -3

How likely do you think these two solutions are? (Hint: not very!)
There is no solution to this quadratic.

 

[galactus]

1. sin?cos? - cos? = 0

If I may add my two cents. As wjm showed for the first, the best thing to do is to factor:

\(\displaystyle cos{\theta}(sin{\theta}-1)=0\)

So, we have \(\displaystyle cos{\theta}=0, \;\ sin{\theta}-1=0\)

\(\displaystyle cos{\theta}=0\)

\(\displaystyle {\theta}=C{\pi}-\frac{\pi}{2}\)

Now, use values of C that fit your constraint of 0 to 2Pi.

i.e., if C=1, we get \(\displaystyle {\theta}=\frac{\pi}{2}\)

This is certainnly betwen 0 and 2Pi.

Likewise for \(\displaystyle sin{\theta}-1=0\)

\(\displaystyle {\theta}=2C{\pi}+\frac{\pi}{2}\)

These are the general solutions for theta. You only need the 0 to 2Pi case.

 

[Deleted member 4993]

2. sec?csc?= csc?

Please help.. you have no idea what it means to me for your help. I greatly appreciate any ideas..

\(\displaystyle sec(\theta)\cdot csc(\theta) - csc(\theta) = 0\)

\(\displaystyle csc(\theta)\cdot [sec(\theta)- 1] = 0\)

\(\displaystyle csec(\theta) = 0 \rightarrow \text no \, solution\)

\(\displaystyle [sec(\theta)- 1] = 0 \rightarrow \theta = 2\cdot n\cdot \pi\)