Trigonometric Questions Dealing with Identities PART I

The following are questions that you may see using your trig identities.

If you do not remember your Trig identities, go to this website:

http://www.sosmath.com/trig/Trig5/trig5/trig5.html 

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EXAMPLE 1.a

What is the value of 1-sin²(2x) if cos²(x)=1 in the domain [0,π]?

a.  1-sin²(2x)=cos²(2x)

You could use the half angle formula, but cos²(x)=1 when x=0 and that is the only time.  If you double the angle 0, then it will still be zero, so the answer is 1.

Example 1.b

Without the use of a calculator, find the value of -(sinx+cosx)(sinx-cosx) given that x=15 degrees.

a. Expand and you get -(sin²x-cos²x)=-sin²x+cos²x=cos(2x)=cos(2(15))=cos(30)= square root of three divided by 2 

sqrt(3)/2

Example 2.a

Note: √ represents square root 

Given √(1-sin²x)(2/√(1+cot²x)), what is the value of this expression when x=15 degrees (no calculator allowed).

square root of all(1-sin²x) and square root of all (1+cot²x) 

a. (1-sin²x)=cos²x, square root of this is simply cosx 

b. (1+cot²x)=cscx, 2/cscx=2sinx because cscx=1/sinx and 1/(1/sinx)=sinx

Now the product is 2sinxcosx=sin2x=sin(2(15))=sin30=1/2