Please help...I'm not understanding trig identities and how to manipulate and express these two problems in their associated functions.
Thanks
a)Express as a function of cos (theta)
2 sin^2(theta) - 1
b)Express as a function of sin (theta) or cos (theta)
tan^2(theta) - 2 sec(theta)sin(theta)
I will use x instead of theta for easier typing
2 sin^2 x - 1 = cos 2x
do you have to prove this?
if so, then
RS = -cos(2x)
= -cos(x+x)
= -cosxcosx + sinxsinx
= -cos^x + sin^x
= -(1-sin^x_ + sin^x
= 2sin^x - 1
= LS
for the second
tan^x - 2secxsinx
= sin^2 x/cos^2 x - 2sinx/cosx
= sin^2 x/cos^2 x 2 sinxcosx/cos^2 x
= (sin^2 x - sin 2x)/(1 - sin^2 x)
this is in terms of sinx
Sure! I can help you with that.
a) To express 2sin^2(theta) - 1 as a function of cos(theta), we can make use of the Pythagorean identity: sin^2(theta) + cos^2(theta) = 1.
Here's how you can do it step by step:
Step 1: Start with the expression 2sin^2(theta) - 1.
Step 2: Replace sin^2(theta) with 1 - cos^2(theta) using the Pythagorean identity.
Step 3: Substitute the expression in step 2 into the original expression:
2(1 - cos^2(theta)) - 1.
Step 4: Simplify the expression further by distributing the 2:
2 - 2cos^2(theta) - 1.
Step 5: Combine like terms:
-2cos^2(theta) + 1.
So, the expression 2sin^2(theta) - 1 can be expressed as -2cos^2(theta) + 1.
b) To express tan^2(theta) - 2sec(theta)sin(theta) as a function of sin(theta) or cos(theta), we can use the trigonometric identities:
sec(theta) = 1/cos(theta) and tan(theta) = sin(theta)/cos(theta).
Here's how you can go about it:
Step 1: Start with the expression tan^2(theta) - 2sec(theta)sin(theta).
Step 2: Substitute sin(theta)/cos(theta) for tan(theta):
(sin(theta)/cos(theta))^2 - 2sec(theta)sin(theta).
Step 3: Substitute 1/cos(theta) for sec(theta):
(sin(theta)/cos(theta))^2 - 2(1/cos(theta))sin(theta).
Step 4: Simplify the expression further:
(sin^2(theta))/cos^2(theta) - 2sin(theta)/cos(theta).
Step 5: Multiply the terms by cos^2(theta) to clear the denominator:
sin^2(theta) - 2sin(theta)cos(theta).
So, the expression tan^2(theta) - 2sec(theta)sin(theta) can be expressed as sin^2(theta) - 2sin(theta)cos(theta).
I hope this helps! Let me know if you have any more questions.