📏 trigonometry

1. المشكلة: لدينا التعبير \( \text{قتا}135 = \text{قا} \) ونريد إيجاد قيمة \( \text{قا} \). 2. التفسير: \( \text{قتا}135 \) تعني \( \cos 135^\circ \) حيث \( \text{ق} \) ترمز إلى \(

1. The problem is to solve the equation $\sin^2(x) = 0$. 2. Recall that $\sin^2(x) = (\sin(x))^2$, so the equation means $\sin(x) = 0$.

1. The problem is to verify the identity $\cos^2(x) = 1$. 2. Recall the Pythagorean identity: $$\sin^2(x) + \cos^2(x) = 1$$ which holds for all real $x$.

1. **Problem statement:** Given $\cos \theta = \frac{6}{10}$ and $\tan \theta < 0$, find $\sin \theta$, $\tan \theta$, $\csc \theta$, $\sec \theta$, and $\cot \theta$. 2. **Recall

1. **Problem Statement:** Given a right triangle with angle $C = 65^\circ$, side $c = 33$ meters opposite angle $C$, and the right angle at vertex $A$, find the lengths of the hypo

1. **Problem Statement:** Two vertical poles each 18 m high stand at points A and B along a straight horizontal road. A footpath meets the road at B from a point C on the footpath

1. **State the problem:** We are comparing the period and amplitude of two functions: the given function $f$ and the function $g(x) = \sin\left(\frac{\pi}{2} x\right)$.\n\n2. **Ide

1. **Problem Statement:** Prove the identity $$\cos^4\theta + \sin^4\theta = 1 - 2\sin^2\theta \cos^2\theta.$$\n\n2. **Recall the Pythagorean identity:** $$\sin^2\theta + \cos^2\th

1. **Prove the formula for $\tan 3A$:** We want to prove:

1. সমস্যাটি হলো: $\sin\left(n\pi + (-1)^n \frac{\pi}{6}\right)$, যেখানে $n$ একটি ধনাত্মক পূর্ণসংখ্যা। 2. আমরা জানি যে $\sin(a + b) = \sin a \cos b + \cos a \sin b$। এখানে $a = n\pi

1. সমস্যাটি হলো: $\sin\left(n\pi + (-1)^n \frac{\pi}{6}\right)$, যেখানে $n$ ধনাত্মক পূর্ণসংখ্যা। 2. সূত্র: আমরা জানি $\sin(a + b) = \sin a \cos b + \cos a \sin b$ এবং $\sin(n\pi) =

1. প্রথম প্রশ্নে বিভিন্ন কোণের ট্রিগনোমেট্রিক মান নির্ণয় করতে হবে। 2. (i) $\sec 3630^\circ$ নির্ণয়:

1. **Problem Statement:** Prove the trigonometric identity $$\cos(A - B) = \cos A \cos B + \sin A \sin B$$ using the vector method. 2. **Concepts and Formula:** We use the dot prod

1. **Problem Statement:** We need to find the correct ratio for $\cos C$ in a right triangle with vertices A, B, and C, where the right angle is at B. The sides opposite vertices A

1. **Problem Statement:** Determine the values of $a$, $b$, and $c$ in the function $f(x) = a \cos(bx) + c$ for $0^\circ \leq x \leq 360^\circ$, given the graph oscillates between

1. **State the problem:** Given a triangle with side $b=8.5$ inches, side $c=18.4$ inches, and angle $A=53.213^\circ$, solve for the remaining side $a$ and angles $B$ and $C$. 2. *

1. **State the problem:** We are given the ratio \( \frac{\sin 50}{u} = \frac{\sin 55}{x} \) and the value \( u = 11.8 \). We want to find \( x \).

1. **State the problem:** Solve the equation $$\frac{8}{f} = \cos 46^\circ$$ for $f$ and give the answer to 2 decimal places. 2. **Recall the formula and rules:** The equation rela

1. The problem is to calculate $\cos(75^\circ) \times 10$ and round the result to 2 decimal places. 2. Recall the cosine of an angle in degrees can be found using a calculator or t

1. Let's start by evaluating the trigonometric expressions given. 2. (a) Evaluate $\cot 42^\circ \times \sec 105^\circ$ and check if it equals $\csc 15^\circ$.

1. **State the problem:** Given a point $P = (8, -6)$ on the terminal side of angle $\theta$, find $\sin \theta$. 2. **Recall the formula:** For a point $(x, y)$ on the terminal si