📏 trigonometry

1. **Problem Statement:** Simplify the given trigonometric expressions and solve for $\theta$ where $\theta \leq 90^\circ$.\n\n1.1 Simplifications:\n\n1.1.1 Simplify $\sin 45^\circ

1. لنفترض أن \(P_2\) تعني نقطة على منحنى أو معادلة محددة ونريد التحقق من صحة العلاقة \(\cos x + \sin x = 1\).\n2. هذه المعادلة ليست صحيحة لكل \(x\) لأنها تشير إلى جمع دوال مثلثية ب

1. The user asks to solve the equation by substituting $\cos x$ or $\sin x$ with $y$, turning it into a quadratic equation. 2. Let us suppose the original equation is of a form lik

1. The problem is to check if $3\sin 2\theta = 3\sin^2 \theta$. 2. Recall the double-angle identity: $\sin 2\theta = 2\sin \theta \cos \theta$.

1. **State the problem:** Solve the equation $$3\sin^2\theta + 5\sin\theta - 4 = 0$$ for $$0 \leq \theta \leq 360^\circ$$. 2. **Substitute:** Let $$x = \sin\theta$$. The equation b

1. The problem is to simplify and evaluate the expression \(\frac{\cos \theta + \sin \theta}{1 - \tan \theta} \quad \text{and} \quad \frac{\sin \theta + \cos \theta}{1 - \cot \thet

1. Statement of the problem: Simplify the expression $\sin^2\theta - 2\cos\theta + \tfrac{1}{4}$.\n2. Use the Pythagorean identity $\sin^2\theta = 1 - \cos^2\theta$ to rewrite the

1. Problem 1: Given $\cot \alpha + \cot \beta = a$, $\tan \alpha + \tan \beta = b$, and $\alpha + \beta = \theta$, prove that $$\tan \theta = \frac{ab}{a - b}$$

1. We are given that $\tan x = \frac{1}{\sqrt{3}}$ and $0^\circ \leq x \leq 90^\circ$. 2. Recall that $\tan x = \frac{\sin x}{\cos x}$, so we have

1. We are given that $\tan x = \frac{1}{\sqrt{3}}$ with $0^\circ \leq x \leq 90^\circ$. We want to find $\cos x - \sin x$. 2. Recall that $\tan x = \frac{\sin x}{\cos x}$. So, $\fr

1. The problem is to find the value of $\frac{1}{60^\circ}$ and verify the given trigonometric identity $\sin 60^\circ = 2 \sin 30^\circ \cos 30^\circ$. 2. First, find the exact va

1. **State the problem:** Given that $\tan x = \frac{1}{\sqrt{3}}$ for $0^\circ \leq x \leq 90^\circ$, find the value of $\cos x - \sin x$. 2. **Recall the basic trigonometric valu

1. Problem 4 statement: In right triangle ABC the right angle is at C, the hypotenuse AB = $2$ cm, and angle at B = $45^\circ$. 2. Since the triangle is right angled at C and angle

1. Problem 4: Find the lengths marked $a$ and $b$ in triangle $ABC$ where $\angle C=90^\circ$, $AB=2$ cm and $\angle B=45^\circ$. 1. In a right triangle the non-right angles sum to

1. **Calculate the height of a tree whose shadow is 20 m shorter when the sun's elevation angle changes from 30° to 60°**. Let the height of the tree be $h$ meters, the shadow leng

1. Given a right triangle with angles 60\degree, 90\degree, and the sides labeled as follows: side opposite 60\degree angle = $x$, side adjacent to 60\degree angle = 6, and hypoten

1. সমস্যা: যদি $\cot x + \csc x = p$ হয়, তবে দেখাতে হবে যে $$ (p^2+1) \cos x + (p^2+1) \sin x = p + 1 - 2.$$ \n\n2. দেওয়া আছে, $\cot x = \frac{\cos x}{\sin x}$ এবং $\csc x = \fra

1. The most common angle formulas are for right triangles and trigonometric identities. 2. In a right triangle, the sum of angles is $180^\circ$ and one angle is $90^\circ$. So, if

1. **Problem Statement:** Find $\cos 66^\circ$ using the given right-angled triangle. 2. **Recall the definition:** $\cos \theta = \frac{\text{length of adjacent side}}{\text{lengt

1. The problem asks to write \( \cos 66^\circ \) as a fraction using the right-angled triangle provided. 2. Recall that \( \cos \theta = \dfrac{\text{length of adjacent side}}{\tex

1. Stating the problem: Verify if $$\frac{2\csc^2 A - 2\csc A \cot A}{-2\cot^2 A + 2\csc A \cot A} = \sec A$$. 2. Factor the numerator and denominator: