Which of the following are trigonometric identities? Select all that apply (there are 3 answers).
A cos^2(theta)=sin^2(theta)-1
B sin(theta)=1/csc(theta)
C sec(theta)=1/cot(theta)
D cot(theta)=cos/sin(theta)
E 1+cot^2(theta)=csc^2(theta)
There is no textbook for this course.
B, D and E and the correct respones
Well, well, well, let's see which of these trigonometric identities are as funny as my nose!
A. cos^2(theta) = sin^2(theta) - 1: Oh, this one is a classic! It's the Pythagorean identity in disguise! Cosine squared equals sine squared minus one. It always cracks me up how those trig functions play around with each other.
B. sin(theta) = 1/csc(theta): Oh, this one is just too good! It's the reciprocal identity for sine, or should I say, the sin-ciprocal identity! Sine equals one over cosecant. That's a fancy way of saying they're the best of friends!
C. sec(theta) = 1/cot(theta): Ah, the secant and cotangent are at it again, trying to confuse us. This one is the reciprocal identity for secant, my circus-loving friends. Secant equals one over cotangent. Talk about a math sideshow!
D. cot(theta) = cos/sin(theta): Now here's a funny one. Cotangent equals cosine divided by sine. The cot's in town to divide some trig functions! It's like a juggling act, but with numbers.
E. 1 + cot^2(theta) = csc^2(theta): Ah, the Pythagorean identity is pulling off another disguise! This time it's 1 plus cotangent squared equals cosecant squared. It's like they're playing hide and seek with their identities. Tricky, tricky!
So, my dear friend, the trigonometric identities in this list are:
A. cos^2(theta) = sin^2(theta) - 1
B. sin(theta) = 1/csc(theta)
E. 1 + cot^2(theta) = csc^2(theta)
Keep those identities laughing!
To determine which of the given options are trigonometric identities, we need to evaluate each option and check if it holds true for all values of theta.
A. cos^2(theta) = sin^2(theta) -1
This is indeed a trigonometric identity known as the Pythagorean Identity. By using the Pythagorean Theorem, we know that sin^2(theta) + cos^2(theta) = 1. Rearranging this equation, we get cos^2(theta) = 1 - sin^2(theta). Therefore, option A is a trigonometric identity.
B. sin(theta) = 1/csc(theta)
Recall that csc(theta) is the reciprocal of sin(theta). Therefore, this equation can be rewritten as sin(theta) = sin(theta)/1, which is always true. Hence, option B is a trigonometric identity.
C. sec(theta) = 1/cot(theta)
To determine if this equation is a trigonometric identity, we need to express sec(theta) and cot(theta) in terms of sin(theta) and cos(theta).
Using the definitions sec(theta) = 1/cos(theta) and cot(theta) = cos(theta)/sin(theta), we can rewrite the equation as 1/cos(theta) = 1/[cos(theta)/sin(theta)].
Taking the reciprocal of the denominator, we get 1/cos(theta) = sin(theta)/cos(theta). Canceling out cos(theta), we are left with 1 = sin(theta). However, this equation is not true for all values of theta. Therefore, option C is not a trigonometric identity.
D. cot(theta) = cos(theta)/sin(theta)
This equation is indeed a trigonometric identity. By definition, cot(theta) = cos(theta)/sin(theta). Thus, option D is a trigonometric identity.
E. 1 + cot^2(theta) = csc^2(theta)
To see if this equation is a trigonometric identity, we can express cot(theta) and csc(theta) in terms of sin(theta) and cos(theta).
Using the definitions cot(theta) = cos(theta)/sin(theta) and csc(theta) = 1/sin(theta), we can rewrite the equation as 1 + (cos(theta)/sin(theta))^2 = (1/sin(theta))^2.
Simplifying the equation, we get 1 + cos^2(theta)/sin^2(theta) = 1/sin^2(theta).
Multiplying both sides by sin^2(theta), we have sin^2(theta) + cos^2(theta) = 1, which is the Pythagorean Identity. Therefore, option E is a trigonometric identity.
In conclusion, options A, B, and E are trigonometric identities. Options C and D are not trigonometric identities.
But I believe it's B, C and D
Only B is correct in your choices.
What are your resources for this major topic if you don't have a textbook ?
The correct relations are listed in your textbook or in your notes.
They are easy to look up
Let me know which ones you think do apply.