📏 trigonometry

1. **Problem statement:** A surveyor stands at point A. The true bearing of a tree from A is 070°.

1. **Problem:** An airplane flies on a bearing of 120° for 200 km, then turns to a bearing of 210° for 150 km. Calculate its distance from the starting point. 2. **Formula and rule

1. **Problem statement:** Two cars start from the same intersection where roads meet at 34°. The slower car travels at 80 km/h, the faster at 100 km/h. After 2 hours, a helicopter

1. The problem states: Given $\sin 2\theta = \frac{7}{11}$, find the value of $\sin 2\theta$. 2. The formula for the sine of a double angle is $\sin 2\theta = 2 \sin \theta \cos \t

1. **Stating the problem:** Verify if the equation $$\frac{\sin x}{1 + \cos x} + \frac{\cos x}{\sin x} = \csc x$$ is true.

1. **Problem statement:** A boat sails 8 km north from point P to Q, then 6 km west from Q to R. We need to find the bearing of R from P and the distance from P to R. 2. **Formula

1. **State the problem:** Solve the trigonometric equation $$2\cos^2(x) - 3\cos(x) + 1 = 0$$ for $$x$$ in the interval $$[0, 2\pi]$$. 2. **Identify the formula and substitution:**

1. **State the problem:** Solve the trigonometric equation $$2\cos^2 x - 3\cos x = 0$$ for $x$. 2. **Formula and rules:** This is a quadratic equation in terms of $\cos x$. We can

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

1. **Problem statement:** We have two tracking stations A and B on the ground, 50 miles apart horizontally. The angles of elevation to a satellite from A and B are 87.0° and 84.2°,

1. **Problem 1: Solve triangle ABC with given $A=38^\circ40'$ , $a=9.72$ m, and $b=11.8$ m.** 2. Use the Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B}$$

1. The problem is to provide the formulas for SOHCAHTOA, the sine law, and the cosine law. 2. SOHCAHTOA is a mnemonic to remember the definitions of sine, cosine, and tangent in a

1. The problem is to solve the inequality $\cos x > -\frac{1}{2}$. 2. Recall that the cosine function has a range of $[-1,1]$ and is periodic with period $2\pi$. The inequality ask

1. The problem is to find the values of $x$ for which $\cos x > -\frac{\sqrt{2}}{2}$.\n\n2. Recall that $\cos x$ ranges between $-1$ and $1$. The value $-\frac{\sqrt{2}}{2}$ is app

1. The problem is to solve the inequality $\sin x < \frac{\sqrt{2}}{2}$.\n\n2. Recall that $\sin x$ is the sine function, which oscillates between $-1$ and $1$. The value $\frac{\s

1. The problem asks for the expression of $\tan 4\theta$ in terms of $\tan \theta$ or $\tan 2\theta$. 2. We use the double angle formula for tangent: $$\tan 2\alpha = \frac{2 \tan

1. Problem: Find the value of $\sqrt{4\sin^2 \left(\frac{\pi}{24}\right)}$. 2. Formula and rules: Recall that $\sqrt{a^2} = |a|$, so

1. The problem states: Given $\cos \theta = \frac{1}{2} \left(a + \frac{1}{a}\right)$, find the value of $\cos 5\theta$. 2. We use the multiple-angle formula for cosine: $$\cos 5\t

1. The problem is to simplify or work with the expression involving $\tan^2 A$, not $\tan 2A$. 2. Recall the identity for $\tan^2 A$: it is simply the square of $\tan A$, i.e., $\t

1. **Problem Statement:** Prove that $$\tan 2A \cdot \sec^2 (90^\circ - A) - \sin^2 A \cdot \csc^2 (90^\circ - A) = 1$$ 2. **Recall the formulas and identities:**

1. **Problem Statement:** Prove the following trigonometric identities given angles $A=0^\circ$, $B=30^\circ$, $C=45^\circ$, $D=60^\circ$, and $E=90^\circ$: