📏 trigonometry

1. **State the problem:** We are given the function $$y = -1 + \frac{3}{2} \cos(2x - \pi)$$ and asked to understand its graph shape, including midline, amplitude, period, phase shi

1. **State the problem:** We are given a triangle with three angles: $\frac{\pi}{7}$, $\frac{5\pi}{14}$, and $\theta$. We need to find the exact value of $\theta$. 2. **Recall the

1. The problem asks to fill in the sine and cosine values for the angles $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$ without looking. 2. Recall the key trigonomet

1. **Problemos teiginys:** Įrodyti, kad $$\cos \alpha - \sin (150^\circ + \alpha) - \sin (150^\circ - \alpha) = 0.$$

1. **State the problem:** Solve the equation $$\sin x + \cos x + \sin x \cos x = 1$$.

1. **State the problem:** Solve the equation $$\sin x + \cos x + \sin x \cos x = 1$$. 2. **Use the identity:** Recall that $$(\sin x + \cos x)^2 = \sin^2 x + \cos^2 x + 2 \sin x \c

1. **State the problem:** We need to find the length of a telephone pole that is tilted $8^\circ$ away from the sun when the sun's elevation angle is $62^\circ$ and the shadow leng

1. **State the problem:** Find the angle $A$ such that $\tan A = \frac{73.7}{39.4}$. 2. **Formula used:** To find the angle from a tangent value, use the inverse tangent function:

1. **State the problem:** Calculate the value of the expression $$\frac{100 \tan 24^\circ}{\tan 45^\circ 50' - \tan 24^\circ} + \tan 45^\circ 50'$$.

1. **State the problem:** We want to simplify the expression $$\sec 2x - \tan 2x$$ for $0 < x < \frac{\pi}{4}$ and identify which of the given options it equals. 2. **Recall the de

1. **State the problem:** We need to find the length $c$ in triangle CBA where $b = 15$, angle $C = 20^\circ$, and angle $B = 125^\circ$. 2. **Find angle $A$:** The sum of angles i

1. **State the problem:** We are given a triangle with angles $A = 64^\circ$, $B = 50^\circ$, and side $b = 17$ opposite angle $B$. We need to find side $a$ opposite angle $A$.\n\n

1. **State the problem:** We are given a right triangle with angle $B = 17.2^\circ$ and side $b = 33.5$ (adjacent to angle $B$). We need to find angle $A$, side $a$, and hypotenuse

1. **State the problem:** We need to find an equation of the form $y = a \sin bx$ or $y = a \cos bx$ that matches the given sinusoidal graph. 2. **Identify key features:**

1. **State the problem:** Prove the identity $$(\sin \theta + \cos \theta)^2 = 1 + 2 \sin \theta \times \cos \theta$$. 2. **Recall the formula:** The square of a sum is given by $$

1. **State the problem:** Find angles with the same terminal side as given angles within specified ranges. 2. **Recall the formula:** Angles that differ by multiples of $360^\circ$

1. **State the problem:** We need to position five points on the graph of the same sine function $y = A\sin(x)$ without shifting along the x-axis, such that no more than two points

1. **Stating the problem:** Understand the basic concepts of trigonometric functions. 2. **Definition:** Trigonometric functions relate the angles of a right triangle to the ratios

1. The user asked about trigonometry graphs. 2. Trigonometric functions like sine, cosine, and tangent are periodic and have characteristic wave-like graphs.

1. **Problem Statement:** Learn the basics of grade 12 trigonometry, including definitions, formulas, and applications. 2. **Key Concepts:** Trigonometry studies relationships betw

1. **Check problem ①:** Given point $P(-5,12)$, calculate radius $r$ using the distance formula: $$r = \sqrt{x^2 + y^2} = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$