SOLUTION: I have three questions that I need extreme help on. 1. Find sec thata if sin theta=-4/5 and 270 degrees<theta<360 degrees. 2. Simplify: (1/sec theta + sin^2 theta/cos theta)cos

Algebra ->  Trigonometry-basics -> SOLUTION: I have three questions that I need extreme help on. 1. Find sec thata if sin theta=-4/5 and 270 degrees<theta<360 degrees. 2. Simplify: (1/sec theta + sin^2 theta/cos theta)cos      Log On


   

Question 172974: I have three questions that I need extreme help on.
1. Find sec thata if sin theta=-4/5 and 270 degrees 2. Simplify: (1/sec theta + sin^2 theta/cos theta)cos theta.
3. Find the exact value of: cos315 degrees.
I have to show my work to get any credit. Please help.
Thanks
Found 2 solutions by midwood_trail, solver91311:
Answer by midwood_trail(310) About Me  (Show Source):
You can put this solution on YOUR website!
I will do number 3 and leave the rest to others.
3. Find the exact value of: cos315 degrees.

Cos315 lies in quadrant 4.
We need to find its reference angle.
To do so, subtract 315 degrees from 360 degrees.
Then:
360 degrees - 315 degrees = 45 degrees
The cosine of an angle is positive in quadrant 4.
The exact value of cos315 = cos45 = sqrt{2}/2
Final answer: sqrt{2}/2

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
Note: I use x instead of theta because theta won't render on this system.

sin%28x%29=sqrt%28cos%5E2%28x%29-1%29

sqrt%28cos%5E2%28x%29-1%29+=+-4%2F5

cos%5E2%28x%29-1=16%2F25

cos%5E2%28x%29=16%2F25%2B1=41%2F25

cos%28x%29=sqrt%2841%29%2F5 or cos%28x%29=-sqrt%2841%29%2F5

sec%28x%29=1%2Fcos%28x%29 so sec%28x%29=5%2Fsqrt%2841%29 or sec%28x%29=-5%2Fsqrt%2841%29 but you need to rationalize the denominators so sec%28x%29=%285sqrt%2841%29%29%2F41 or sec%28x%29=-%285sqrt%2841%29%29%2F41

***************

2. %281%2Fsec%28x%29+%2B+sin%5E2%28x%29%2Fcos%28x%29%29cos%28x%29

1%2Fsec%28x%29=cos%28x%29 so:

%28cos%28x%29+%2B+sin%5E2%28x%29%2Fcos%28x%29%29cos%28x%29

Apply LCD of cos%28x%29

%28%28cos%5E2%28x%29+%2B+sin%5E2%28x%29%29%2Fcos%28x%29%29cos%28x%29

But cos%5E2%28x%29%2Bsin%5E2%28x%29=1 so:

%281%2Fcos%28x%29%29cos%28x%29=1

********************

3. An angle of 315° is equivalent to an angle of -45° either of which is formed by a ray bisecting the right angle formed by the positive x-axis and the negative y-axis. This ray intersects the unit circle at a point that forms the vertex of an isoceles right triangle, the other two vertices being the origin and the point (cos%28x%29,1) and having a hypotenuse of length 1. The Pythagorean theorem gives us that the legs of the triangle are of length sqrt%282%29%2F2. The x coordiate of the vertex at the unit circle is positive and the y coordinate is negative because this is a Quadrant IV angle. Hence, cos%28315%29=%28sqrt%282%29%2F2%29%2F1=sqrt%282%29%2F2