📏 trigonometry

1. Problem 4: In triangle $\triangle ABC$, given $B=35^\circ$, $C=40^\circ$, and side $a=12$ cm, find side $c$. 2. Use the Law of Sines: $$\frac{a}{\sin A} = \frac{c}{\sin C}$$

1. Problem: Solve $2 \sin \theta - \sqrt{2} = 0$ for $0^\circ \leq \theta \leq 360^\circ$. Formula: $\sin \theta = \frac{\sqrt{2}}{2}$.

1. State the problem. In triangle ABC we are given $b=4.2\,\text{cm}$, $c=7.5\,\text{cm}$ and $A=48^\circ36'$.

1. The problem involves evaluating multiple trigonometric expressions and identities. 2. Recall key trigonometric identities:

1. **Problem Statement:** We have two functions: $f(x) = -2 \cos x$ and $g(x) = \sin 2x$ defined on the interval $-90^\circ \leq x \leq 180^\circ$. We need to:

1. **Stel die probleem:** Ons het twee funksies: $$f(x) = a \sin(bx) + 1$$

1. **Problem:** Find all values of $\theta$ such that $\sin \theta = \frac{1}{2}$ and $0 \leq \theta \leq 2\pi$. 2. **Formula and rules:** The sine function gives the ratio of the

1. **Problem statement:** We have a right triangle with angles 45°, 30°, and 90°, and the base (adjacent to the 45° angle) is 26 cm. We need to find the hypotenuse length $d$. 2. *

1. **Problem Statement:** We are asked to analyze the function $y = \cos x$ for the domain $0^\circ \leq x \leq 360^\circ$.

1. **Problem statement:** We need to find the length of the hypotenuse in a right-angled triangle where one angle is 60° and the side adjacent to this angle is 6 meters. 2. **Formu

1. **Problem Statement:** We are given that angle $x$ is obtuse, meaning $90^\circ < x < 180^\circ$. We need to determine the truth value of the following statements: a) $\sin x >

1. **State the problem:** We need to define the three primary trigonometric ratios in terms of Cartesian coordinates $x$, $y$, and radius $r$.

1. **Problem statement:** Given functions $f(x) = -2\cos x$ and $g(x) = \sin 2x$ for $-90^\circ \leq x \leq 180^\circ$, we need to: - Draw graphs of $f$ and $g$ on the same axes.

1. نبدأ بقراءة السؤال: لدينا مثلث قائم الزاوية في النقطة ب، ونريد إيجاد طول الضلع أ ج. 2. في مثلث قائم الزاوية، نستخدم علاقات الدوال المثلثية بين الزوايا والأضلاع.

1. **Δήλωση του προβλήματος:** Να αποδείξουμε την ταυτότητα

1. **Problem Statement:** Find the value of $\tan 90^\circ + \sin 90^\circ$ without using a table or calculator. 2. **Recall the definitions and values:**

1. **Problem Statement:** Prove that $$\sec^2\theta + \tan^2\theta = 2\tan^2\theta + 1$$. 2. **Recall the Pythagorean identity:** $$\sec^2\theta - \tan^2\theta = 1$$.

1. Problem: Find the values of the given trigonometric expressions without using a table or calculator. 2. Important formulas and values:

1. **State the problem:** Given $m = \sin\theta + \tan(x\theta)$ and $n = \tan(x\theta) - \sin(x\theta)$, find the value of $m^2 - n^2$ such that $(\cdot)^2 = 16mn$. 2. **Recall th

1. **State the problem:** Prove or verify the identity $$\sqrt{\sec^2 A + \csc^2 A} = \tan A + \cot A.$$\n\n2. **Recall definitions and identities:**\n- $\sec A = \frac{1}{\cos A}$

1. **State the problem:** Prove the identity $$\sqrt{x}(\sec^2 A + \csc^2 A) = \tan A + \cot A$$. 2. **Analyze the equation:** The equation involves trigonometric functions and a s