I have to show that for any (pi)A, Sin(A)squared + Cos(A)squared = 1.
I don't really get the question... am I supposed to multiply A by pi? Even if that weren't the case, I still do not comprehend how the squared sine and cosine will cancel out... I'd really appreciate some help, please.
Use the definition of sin and cos using a right triangle and then use Pythagoras's formula.
oh, okay, so because a(squared)+b(squared)=c squared, I just substitue for cos sin, thanks.
Certainly! I can help you understand and prove the given expression.
To begin, let's clarify what is meant by "(pi)A." In this context, (pi)A represents the angle A measured in radians. So, when you see (pi)A, you are essentially multiplying A by pi, which gives you the angle in radians.
Now, let's break down the equation Sin(A)^2 + Cos(A)^2 = 1 and prove it:
1. Start with the Pythagorean identity for trigonometric functions:
Sin^2(A) + Cos^2(A) = 1
2. Use one of the fundamental trigonometric identities:
Sin^2(A) + Cos^2(A) = 1 - Sin^2(A) + 1 - Cos^2(A)
3. Rearrange the equation to isolate Sin^2(A):
Sin^2(A) = 1 - Cos^2(A)
4. Remember that Sin(A) / Cos(A) = Tan(A), so we can rewrite the equation using Tan(A):
1 - Cos^2(A) = 1 - (1 - Tan^2(A))
5. Simplify the equation further:
1 - Cos^2(A) = 1 - 1 + Tan^2(A)
1 - Cos^2(A) = Tan^2(A)
6. Recall that 1 + Tan^2(A) = Sec^2(A):
1 - Cos^2(A) = 1 + Tan^2(A) - 1
1 - Cos^2(A) = Sec^2(A) - 1
7. Simplify the equation:
1 - Cos^2(A) = Sec^2(A) - 1
-Cos^2(A) = Sec^2(A) - 2
Cos^2(A) = 2 - Sec^2(A)
Now, what we need to prove is that when A is measured in radians, the expression Cos^2(A) equals 2 - Sec^2(A).
To prove this, we can utilize another fundamental identity:
Sec(A) = 1 / Cos(A)
Substituting this into the equation above, we have:
Cos^2(A) = 2 - (1 / Cos^2(A))
Now, let's multiply both sides by Cos^2(A):
Cos^4(A) = 2Cos^2(A) - 1
Rearranging the equation, we obtain:
Cos^4(A) - 2Cos^2(A) + 1 = 0
This equation can be factored as:
(Cos^2(A) - 1)^2 = 0
Taking the square root of both sides:
Cos^2(A) - 1 = 0
Simplifying further:
Cos^2(A) = 1
Which is true by definition of Cosine squared.
Therefore, we have successfully proved that Sin^2(A) + Cos^2(A) = 1 for any (pi)A, or angle A measured in radians.