Trigonometric identities

Question

I'm currently studying pre-calculus, and I’m having difficulty grasping the concept of trigonometric identities. Specifically, I struggle with:

Proving identities: I often get stuck when trying to prove identities like

sin²(x) + cos²(x) = 1

or tan(x) = sin(x) / cos(x)

Solving equations: I find it challenging to solve equations involving multiple trigonometric functions, such as

sin(2x)=cos(x).

I would greatly appreciate any recommendations for:

Online resources or videos that explain these concepts clearly and provide practice problems.

Study techniques that can help me memorize these identities and apply them effectively.

Tutoring services that specialize in trigonometry or pre-calculus, where I can receive personalized help.

slope

Answer 1

For proving the basic identities like the Pythagorean identity that you mentioned above, I recommend drawing a unit circle and a triangle on that circle. This helps since the x-coordinate of the triangle will correspond to cos(θ) and the y-coordinate of the triangle will correspond to sin(θ).

Solving equations are difficult! I think the key point to these are trying to make all the trigonometric functions into the same angle and then of the same type.

For example, for your sin(2x) = cos(x), we notice that 2x is not the same angle as x. So, we use the double angle formula to transform sin(2x) into 2sin(x)cos(x). Then, we can divide both sides by cos(x) to finish the problem.

One thing I recommend is to memorize some of the trig identities such as the pythagorean identities:

cos^2(x) + sin^2(x) =1

tan^2(x) + 1 = sec^2(x)

cot^2(x) + 1 = csc^2(x)

some other ones like these:

cos(pi/2 - x) = sin(x)

sin(pi/2 - x) = cos(x)

sin and cos addition formulas:

sin(x + y) = sin(x)cos(y) + sin(y)cos(x)

cos(x + y) = cos(x)cos(y) - sin(x)sin(y)