📏 trigonometry

1. The problem is to evaluate the expression $2 \tan(\sin^{-1}(-1))$. 2. Recall that $\sin^{-1}(x)$ is the inverse sine function, which returns an angle $\theta$ such that $\sin(\t

1. The problem states: Given that $x + y = 90^\circ$, find the value of the expression $$\frac{\cos(2x + y) + \sin(3x + 2y)}{\cos y}$$

1. **Problem:** Given $\sec\theta + \tan\theta = p$, find the value of $\sin\theta$ in terms of $p$. 2. **Formula and rules:** Recall the identity:

1. **State the problem:** Simplify and verify the identity $$(\cot\theta + \csc\theta)(\tan\theta - \sin\theta) = \sec\theta - \cos\theta.$$\n\n2. **Recall definitions and formulas

1. **Problem:** Define the ASTC Rule. 2. **Explanation:** The ASTC Rule is a mnemonic to remember the signs of trigonometric functions in the four quadrants of the unit circle.

1. The problem is to understand or use the angle given, which is $36.9^\circ$. 2. Angles are measured in degrees and are used in various math and physics problems, such as trigonom

1. Problem: Evaluate $\cos(625)$.\n2. Formula and rules: Cosine is periodic with period $2\pi$, so $$\cos(x+2\pi k)=\cos x\text{ for any integer }k$$.\n3. Reduce the angle modulo $

1. **State the problem:** We need to find the value of $k$ in the equation $$\frac{\cot A}{1 + \csc A} - \frac{\cot A}{1 - \csc A} = \frac{k}{\cos A}.$$\n\n2. **Recall definitions

1. **Problem statement:** Find all values of $\theta$ such that $\tan \theta = 2$. 2. **Formula and rules:** The tangent function is periodic with period $\pi$, meaning if $\tan \t

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