📏 trigonometry

1. **Problem statement:** We have two right triangles sharing an angle of 15°. The top-right triangle has hypotenuse $r=75$ and height $x$. We want to find $x$. 2. **Formula used:*

1. The problem is to verify the identity: $$\sin^4 x - \cos^4 x = 1 - 2\cos^2 x$$. 2. Recall the difference of squares formula: $$a^2 - b^2 = (a-b)(a+b)$$.

1. **State the problem:** Prove or verify the identity $$(\sin A + \cos A)^2 + (\sin A - \cos A)^2 = 2.$$\n\n2. **Recall the formula:** The square of a sum and difference are given

1. **Problem statement:** Solve triangle ABC given $A=43^\circ$, $b=7$ cm, and $c=6$ cm. 2. **Known values:**

1. Let's understand where the number 0.707 comes from. 2. The value 0.707 is approximately equal to $\frac{1}{\sqrt{2}}$.

1. We are asked to show that $$\frac{\cot^2 \alpha}{\csc \alpha - 1} = \csc \alpha + 1$$. 2. Recall the identities:

1. Diberikan bahwa $\sin 23^\circ = m$. Kita diminta mencari nilai $\cos 113^\circ$.\n\n2. Gunakan identitas sudut pelengkap dan hubungan antara sinus dan kosinus: $$\cos \theta =

1. The problem is to find the value of $\tan 135^\circ$. 2. Recall the tangent function and its properties: $\tan(\theta) = \frac{\sin \theta}{\cos \theta}$.

1. **State the problem:** We want to show that $$3 \tan^2 \theta + 5 \sin^2 \theta \equiv \frac{8 \sin^2 \theta - 5 \sin^4 \theta}{1 - \sin^2 \theta}$$ for any angle $$\theta$$. 2.

1. **State the problem:** An observer sees two planes at different altitudes and angles of elevation. We need to find the distance between the two planes. 2. **Given:**

1. **Problem:** Solve the equation $\sin(x + \frac{\pi}{6}) + \sin(x - \frac{\pi}{6}) = \frac{1}{2}$ for $x$ in $[0, 2\pi]$. 2. **Formula and rules:** Use the sum-to-product identi

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Recall formulas and identities:**

1. The problem asks which trigonometric identity is NOT correct for a point $(x,y)$ on the unit circle with rotation angle $\theta$. 2. Recall the unit circle definitions:

1. **Problem statement:** Given quadrilateral ABCD with sides BC = 192 m, CD = 287.9 m, BD = 168 m, and AD = 205.8 m, we need to: (a)(i) Calculate angle CBD and show it rounds to 1

1. **State the problem:** We need to find the value of $\tan(\arccos(\frac{5}{13}) + \arcsin(\frac{3}{5}))$. 2. **Recall the formula for tangent of a sum:**

1. **State the problem:** We are given the function $y=4\cos\left(2x+\frac{2\pi}{3}\right)-1$ and want to understand its properties. 2. **Formula and explanation:** The function is

1. **Problem Statement:** Prove the trigonometric identity: $$\frac{\cos x + 1}{\sin^2 x} = \frac{\csc x}{1 - \cos x}$$

1. **Problem:** Solve the equation $\cos(6x) + 3 \sin(3x) = 2$ for $0 \leq x < 2\pi$. 2. **Recall the range of trigonometric functions:**

1. **Problem Statement:** Find the exact value of $\cos\left(\frac{13\pi}{12}\right)$.\n\n2. **Formula and Rules:** We use the cosine addition formula: $$\cos(a+b) = \cos a \cos b

1. **State the problem:** Find the related acute angle of $\frac{14\pi}{3}$. Related acute angle means the smallest positive angle between the terminal side of the given angle and

1. **State the problem:** Prove the identity $$35 \sec x - \tan x = \tan\left(\frac{\pi}{4} - \frac{x}{2}\right)$$. 2. **Recall relevant formulas:**