📏 trigonometry

1. **State the problem:** Simplify the expression $2 - 2\sin A - 2\sin A \cos A + 2\cos A$. 2. **Group like terms:** Group terms involving $\sin A$ and $\cos A$ separately:

1. **State the problem:** Verify the identity $$(1 - \sin A + \cos A)^2 = 2(1 - \sin A)(1 + \cos A)$$. 2. **Expand the left side:** Use the formula for squaring a trinomial $$(a +

1. **State the problem:** Verify the trigonometric identity $1 - \sin A + \cos A = 2(1 - \sin A)(1 + \cos A)$.\n\n2. **Recall the formula and rules:** We will expand the right-hand

1. **State the problem:** Simplify the expression $2(1 - \sin A)(\cos A)$. 2. **Recall the distributive property:** $a(b+c) = ab + ac$. We will apply this to expand the product.

1. **State the problem:** Given that $\csc A = \frac{13}{12}$, find the value of $$\frac{2 \sin A - 3 \cos A}{4 \sin A - 9 \cos A}.$$\n\n2. **Recall the definitions and formulas:**

1. مسئله را بیان می‌کنیم: اگر $x$ در ربع دوم دایره مثلثاتی باشد و روابط $x = 1 + \sin x$ و $\cos^2 x$ برقرار باشد، مقدار عبارت $$x^2 \tan^2 x - \cot^2 x$$ را بیابید. 2. ابتدا باید

1. **State the problem:** Simplify the expression $$\frac{1-\sin x}{\cos x}$$. 2. **Recall the Pythagorean identity:** $$\sin^2 x + \cos^2 x = 1$$.

1. **State the problem:** Prove the identity $1 + \csc\theta = \csc\theta (1 + \sin\theta)$. 2. **Recall definitions and formulas:** Recall that $\csc\theta = \frac{1}{\sin\theta}$

1. **Problem:** Find the value of $\sec(\arctan(\frac{2}{3}))$. 2. **Formula and rules:** Recall that $\sec(\theta) = \frac{1}{\cos(\theta)}$ and $\arctan(x)$ gives an angle $\thet

1. **Problem:** Find $x$ in triangle (a) with angles 85°, 45°, and side 9cm opposite 85°. 2. **Step 1:** Use the fact that the sum of angles in a triangle is 180°.

1. The problem is to find the value of $\sin\left(\frac{\pi}{2}\right)$.\n\n2. The sine function, $\sin(\theta)$, gives the y-coordinate of a point on the unit circle at an angle $

1. The problem is to understand the concept of radians and how to work with angles measured in radians. 2. Radians measure angles based on the radius of a circle. One radian is the

1. **Problem statement:** From a viewing tower 30 m above the ground, the angle of depression to an object on the ground is 36°. The angle of elevation to an aircraft vertically ab

1. The problem asks for the definitions of the tangent and cosine functions in trigonometry. 2. The tangent of an angle in a right triangle is defined as the ratio of the length of

1. **State the problem:** Find $\sin(B)$ for a right triangle where the vertical side is $\sqrt{13}$, the horizontal side is 6, and the hypotenuse is 7. Angle $B$ is between the ve

1. The problem is to prove the identity $$\arctan(1) + \arctan(2) + \arctan(3) = \pi$$. 2. We use the formula for the sum of arctangents:

1. The problem asks to find the values of $\sin 135^\circ$ and $\tan 15^\circ$.\n\n2. Recall the formulas and identities:\n- $\sin(180^\circ - \theta) = \sin \theta$\n- $\tan(45^\c

1. **نص المسألة:** حل المعادلة المثلثية $$2 \sin 2\theta + \sin \theta = 1$$ حيث \(\theta\) بالدرجات. 2. **القوانين المستخدمة:**

1. **Problem statement:** We need to find the height of a tree given two points A and B on the ground, 30 m apart, with a right angle (90°) between them at the tree's base. The ang

1. The problem asks to find the angle $\theta$ in radians such that $\sin \theta = \frac{\sqrt{2}}{2}$. 2. Recall that on the unit circle, the sine of an angle $\theta$ corresponds

1. The problem is to find the value of $\sin 240^\circ$ using the unit circle. 2. Recall that the unit circle defines sine as the y-coordinate of the point on the circle at a given