📏 trigonometry

1. **Problem Statement:** Prove that $$\frac{1}{2} \cos^{-1} x = \cos^{-1} \sqrt{\frac{1 + x}{2}}.$$\n\n2. **Recall the double-angle formula for cosine:** $$\cos 2\theta = 2\cos^2

1. Problem: Factorize and simplify the expression \(\cos 2x + \cos 4x + \cos 6x\). 2. Use the sum-to-product formulas: \(\cos A + \cos B = 2 \cos \frac{A+B}{2} \cos \frac{A-B}{2}\)

1. Stating the problem: Factorize and simplify the expressions: (a) $$\frac{\sin x + \cos\left(x - \frac{\pi}{6}\right)}{\sin x + \sin\left(x - \frac{2\pi}{3}\right)}$$

1. The problem is to solve a triangle using the sine rule and cosine rule without using the semiperimeter. 2. The sine rule states that for any triangle with sides $a$, $b$, $c$ an

1. **State the problem:** Verify the trigonometric identity $$\cot \beta + \tan \beta = \sec \beta \csc \beta$$. 2. **Recall definitions and formulas:**

1. **State the problem:** Solve the trigonometric equation $$2\sin^2(x) + \sin(x) - 1 = 0$$ and find the general solutions in radians. 2. **Use substitution:** Let $$y = \sin(x)$$.

1. The problem is to match the given trigonometric formulas/identities (A-L) with the numbered formulas 1 to 9. 2. We start by identifying each formula/identity letter with its cor

1. The problem asks whether the amplitude of a trigonometric graph can be negative. 2. The amplitude of a trigonometric function like $y = A \sin(x)$ or $y = A \cos(x)$ is defined

1. **Problem statement:** Sketch the graph of the function $$y = 2\sin x - \tan x$$. 2. **Formula and important rules:**

1. The problem asks for the value of $\arctan(1)$, which is the angle whose tangent is 1. 2. Recall the definition: $\arctan(x)$ is the inverse function of $\tan(\theta)$, so $\arc

1. The problem is to simplify $\sin\left(\frac{3\pi}{2} - x\right)$.\n\n2. We use the sine difference identity: $\sin(a - b) = \sin a \cos b - \cos a \sin b$.\n\n3. Applying this i

1. The problem is to simplify $\sin\left(\frac{3\pi}{2} - x\right)$. 2. We use the sine difference identity:

1. The problem is to simplify $\sin\left(\frac{3\pi}{2} - x\right)$. 2. We use the sine difference identity:

1. **State the problem:** Simplify the expression $\sin\left(\frac{3\pi}{2} - x\right)$. 2. **Recall the sine difference identity:** For any angles $A$ and $B$,

1. مسئله: دنباله $a_n = \frac{1}{\sin n}$ را برای $n=45060$ پیدا کنید. 2. ابتدا باید مقدار $\sin 45060$ را محاسبه کنیم. چون سینوس تابعی تناوبی با دوره $360^\circ$ است، می‌توانیم $4

1. مسئله: دنباله $a_n = \frac{1}{\sin n}$ را برای $n=4560$ بیابید. 2. فرمول و نکات مهم: برای محاسبه مقدار دنباله، باید مقدار $\sin 4560$ را بیابیم و سپس معکوس آن را محاسبه کنیم.

1. **State the problem:** Solve the trigonometric equation $$\cos 2x - \cos x + 1 = 0$$. 2. **Recall the double-angle formula:** $$\cos 2x = 2\cos^2 x - 1$$.

1. **State the problem:** Solve the trigonometric equation $$\cos 2x - \cos x - 1 = 0$$. 2. **Recall the double-angle formula:** $$\cos 2x = 2\cos^2 x - 1$$.

1. **State the problem:** Evaluate the expression $\tan 1^2 - 1$. 2. **Clarify the expression:** The expression can be interpreted as $\tan(1^2) - 1$, which means take the tangent

1. Problem: Write each expression as a trigonometric function of a single angle. (i) sin 37° cos 22° + cos 37° sin 22°

1. **State the problem:** Prove that $$\frac{\sin A + \cos A - 1}{\sin A - \cos A + 1} = \frac{\cos A}{1 + \sin A}$$. 2. **Start with the left-hand side (LHS):** $$\frac{\sin A + \