📏 trigonometry

1. **State the problem:** Solve the equation $$4 \tan \theta + 5 \sin \theta = 0$$ for $$0 < \theta \leq 360^\circ$$. 2. **Recall the definitions and formulas:**

1. **State the problem:** We need to sketch the graphs of $y = \sin 2x$ and $y = 1 + \cos 2x$ for $0^\circ \leq x \leq 360^\circ$ and find the number of solutions to the equation $

1. **State the problem:** Prove the identity $$\frac{\sin x \tan x}{1-\cos x} \equiv 1 + \frac{1}{\cos x}$$ and then solve the equation $$\frac{\sin x \tan x}{1-\cos x} + 2 = 0$$ f

1. **State the problem:** We need to find the general solution for the trigonometric equation $$\cos(2x) = \sin(x)$$. 2. **Recall the double-angle formula:** $$\cos(2x) = 1 - 2\sin

1. We are asked to solve the equation $2\sin^2 x + \sin x - 1 = 0$ for $0^\circ \leq x \leq 360^\circ$. 2. Use the substitution $u = \sin x$ to rewrite the equation as a quadratic:

1. مسئلہ بیان کریں: مثلث ABC میں، جہاں زاویہ B قائمہ ہے، ہمیں زاویہ X کے لیے سائن اور کاز معلوم کرنا ہے۔ 2. فارمولا اور اصول: مثلث قائمہ الزاویہ میں، پائتھاگرین تھیورم استعمال ہوتا

1. **Problem:** A building casts a shadow of 110 feet. The angle of elevation of the top of the building from the tip of the shadow is 29°. Find the height of the building. 2. **Fo

1. **State the problem:** We need to write a sine function with the following characteristics: - Midline (vertical shift) at $y=3$

1. **Problem statement:** From a tower 32 m high, a car is observed at an angle of depression of 55 degrees. We need to find the horizontal distance of the car from the base of the

1. **State the problem:** Prove the trigonometric identity $$1 - \cos 5\theta \cos 3\theta - \sin 5\theta \sin 3\theta = 2 \sin^2 \theta$$. 2. **Recall the cosine addition formula:

1. The problem is to verify the identity $$\frac{2\tan\theta}{1+\tan^2\theta} = \sin 2\theta$$. 2. Recall the double-angle formula for sine: $$\sin 2\theta = 2\sin\theta\cos\theta$

1. **State the problem:** Simplify and verify the identity: $$\frac{\cos^2 a + \cot a}{\cos^2 a - \cot a} = \frac{\cos^2 a \tan a + 1}{\cos^2 a \tan a - 1}$$

1. **State the problem:** We want to graph one period of the function $$y = -3 \cos\left(\frac{1}{2}(x - \frac{\pi}{3})\right) + 1$$ and understand how to move the graph horizontal

1. **Problem statement:** (ক) Given that $\sec A - \tan A = \frac{2}{5}$, find the value of $\sec A + \tan A$.

1. **Problem statement:** Given $\sec A - \tan A = \frac{5}{2}$, find the value of $\sec A + \tan A$. 2. **Formula and important rule:** Use the identity:

1. **State the problem:** We need to solve the equation $\tan(90^\circ - \theta) = \frac{5}{3}$ for $\theta$. 2. **Recall the formula and identity:** The tangent of the complement

1. The problem asks why the angle in the third quadrant is expressed as $180^\circ + \alpha$ instead of $\alpha - 180^\circ$. 2. In trigonometry, angles are measured from the posit

1. **State the problem:** Solve the equation $$\sqrt{10}\sin(\theta - 71.56) = -2$$ for $$\theta$$ in the interval $$0 \leq \theta \leq 360$$ degrees. 2. **Analyze the equation:**

1. **State the problem:** Solve the equation $$\sqrt{10}\sin(\theta - 71.56) = -2$$ for $\theta$. 2. **Recall the range of sine function:** The sine function $\sin(x)$ always satis

1. نبدأ ببيان المشكلة: لدينا مثلث قائم الزاوية بزوايا أ، ب، ج، ونريد إيجاد قيمة \(\sin(A+B)\). 2. في مثلث قائم الزاوية، مجموع الزوايا الداخلية يساوي 180 درجة، وزاوية واحدة تساوي 90

1. **Problem:** Evaluate $\sin^2(60^\circ) + \cos^2(60^\circ)$. 2. **Formula and rule:** The Pythagorean identity states that for any angle $\theta$,