Articles of trigonometry

Proof of trigonometric identity $\sin(A+B)=\sin A\cos B + \cos A\sin B$

All the proofs I’ve seen are geometrical, assuming that $A+B$ is less than $90$ degrees. How can you prove this identity for $A+B$ greater than $90$ degrees, or more generally, any arbitrary value?

summation of a trigonometric series

How to evaluate $\tan^2(1) + \tan^2(3) + \tan^2(5) + \tan^2(7) + \ldots + \tan^2(89)$? Angles are given in degrees. I tried converting $\tan(89)$ as $\cot(1)$ and then tried combining $\tan(1)$ and $\cot(1)$, but later got stuck.

Proof of a trigonometric expression

Let $f(x) = (\sin \frac{πx}{7})^{-1}$. Prove that $f(3) + f(2) = f(1)$. This is another trig question, which I cannot get how to start with. Sum to product identities also did not work.

Proof of $\limsup\sin nx=1, n\rightarrow \infty \forall x\in \mathbb{R}.$

I have a problem: need to prove $$\limsup_{n\to\infty}\sin nx=1$$ for all x without set which measure is equal zero. We need to come up with a sequence with limit that equals to one, but I don’t know how to do this.

Prove that $-3/2\leq\cos a + \cos b + \cos c\leq 3$?

Given 3 non-null vectors $v,u,w$ and angles $a=(u,v), b=(u,w), c=(v,w)$, Prove that $-3/2\leq\cos a + \cos b + \cos c\leq 3$. I’ve managed to prove that: $\cos a + \cos b + \cos c\leq 3$ basically arguing that $\cos \theta$ is bounded by $-1,1$ using the inequality of Cauchy-Schwarz. I reasoned that as the maximum […]

Prove $\cos^2x\sin^4x = \frac{1}{32}(2-\cos(2x)-2\cos(4x)+\cos(6x))$

I need help with a problem some may consider odd but here it is. I have the following trig identity I been working on and I managed to get it to. $$\cos^2x\sin^4x=\frac{3}{16}-\frac{\cos(2x)}{4}+\frac{3\cos(2x)}{16}+\frac{1}{8}(1+\cos(4x))+\frac{1}{32}(\cos(6x)+\cos(2x))$$ However I am not sure how to make this simplify to $\frac{1}{32}(2-\cos(2x)-2\cos(4x)+\cos(6x))=\cos^2x\sin^4x$ and thus I am stuck.

How do I get $\cos{\theta} \lt \frac{\sin{\theta}}{\theta} \lt 1$?

How do I get: $$\cos{\theta} \lt \frac{\sin{\theta}}{\theta} \lt 1$$

prove that : cosx.cos(x-60).cos(x+60)= (1/4)cos3x

I should prove this trigonometric identity. I think I should get to this point : $\cos(3x) = 4\cos^3 x – 3\cos x $ But I don’t have any idea how to do it (I tried solving $\cos(x+60^{\circ})\cos(x-60^{\circ})$ but I got nothing)

How to find the sine of an angle

How to find the sine/cos/tangent/cotangent/cossec/sec of an angle: In degrees $\sin(23^{\circ}) =$ ? In radians $\sin(0.53) =$ ?

Replicating a cosine graph with sine, given transformations?

Say I have a cos function, $y = 1.2cos(0.503x) + 5$ And say I want to replicate it using sine, so expressing the above function in sine so that it gives “the same wave”. How would I do this? I know I have to find a phase shift, but I don’t get the steps on […]