SOLUTION: I think my problem with these kinds of questions is that I�m having a hard time manipulating the secant, cosine, and so on. For instance, when trying to find the exact value for �s

Question 241174: I think my problem with these kinds of questions is that I�m having a hard time manipulating the secant, cosine, and so on. For instance, when trying to find the exact value for �sec 5π/12�; I know that sec=1/cos and I believe that I�m pretty much set up to use a sum difference identity but don�t think that I can just ...safetly assume that 1/cos=cos and I don�t know if I can honestly say how to change 1/cos into cos so I can use the identity. Also, I want to try my best to finish the rest on my own (that�s why I picked only one question). If anyone has any tips on handling Trig indentities and Trigonometry in general, I�d appreciate the advice. Thanks! ^_^
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Whenever a Trig problem refers to "exact values" you will end up using angles that are multiples of 30 or 45 degrees (or or radians).

Since you are looking for the exact value of we have to find a way to express in terms of multiples of and/or . This expression can be a sum or a difference. The expression can be twice or half a special angle. And this expression can be some combination of any of these.

So what expression of multiples of and/or equals ? There are probably others but the one that is most obvious to me is: . SO we will be using half-angle formulas.

There is no half-angle formula for sec, so we will find and then find its reciprocal to get the sec:

(The formula has a "+-" in front of the square root. However, since , is in the first quadrant. And since is in the first quadrant where all Trig functions are positive, we will discard the "-" and use the positive square root.)

Now we just find , substitute and simplify. is in the second quadrant where cos is negative. And its reference angle is . Together this makes . Substituting this into our equation we get:

To simplify this we will start by eliminating the fraction within a fraction. We can do this my multiplying the numerator and denominator of the "big" fraction by the LCD of the "little" fraction(s). Since there is only one "little" fraction, we will use its denominator, 2:

which simplifies to:

Using the property, :

which simplifies to:

Now we can find the sec by finding the reciprocal of cos:

This may be an acceptable answer. However, the denominator is not rational. If you want rationalized denominators:

I'll leave it up to you to multiply this out.

RELATED QUESTIONS
The square of an integer is equal to twice the integer. Find the integer. I am havin a (answered by scott8148)

I do not understand simplifying radicals. Could you please explain to me how to solve... (answered by edjones,josmiceli)

This is my first time submitting anything, so bare with me and let me no if I am doing... (answered by jim_thompson5910)

I'm having a hard time with this problem: {{{2/y}}}+{{{5/2}}}=4-{{{2/3y}}} my book... (answered by bonster)

What is the inverse element of 7 under the binary operation "x" defined on the set of... (answered by CubeyThePenguin)

if tangent of theta equals 5 divided by 7, then cosine of theta will equal ? These types (answered by stanbon)

Hello, I am having a hard time understanding how to graph the linear inequalities. I... (answered by josgarithmetic)

I am having a rather hard time understanding what to do on double-sided equations, like... (answered by solver91311)

Give an example of a real number that is not a rational number.____________ Write... (answered by psbhowmick)