📏 trigonometry

1. **Problem Statement:** Kurt wants to sail from a marina to an island 15 km due east. He sails first on a heading of N 70° E, then on a heading of 120°. We need to find the total

1. The problem states the trigonometric identity: $$\sin^2 \theta + \cos^2 \theta = 1$$ for any angle $\theta$. 2. This is a fundamental Pythagorean identity in trigonometry.

1. **Problem a:** Find the amplitude, period, and phase shift for $y = 2 \cos(3x + \frac{\pi}{2})$ and sketch the graph for $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$. 2. **Amplitu

1. **Énoncé du problème :** Vérifier que pour $t \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, on a $$\sin^2 t = \frac{\tan^2 t}{1 + \tan^2 t}$$

1. **Problem 7:** A flagpole creates a 7.9 m long shadow when the sun is at an angle of elevation of 59°. Find the height of the flagpole. 2. We use the tangent function from SOH C

1. **Determine angle C in the triangle with sides 5 and 13** Given: \( \cos \Phi = \frac{a}{h} = \frac{5}{13} \)

1. The question "Why can't you use cos?" is a bit general, so let's clarify when and why cosine (cos) might not be used in certain math or physics problems. 2. Cosine is a trigonom

1. **State the problem:** We have a right triangle with a right angle at vertex B, side CB = 30 mm, angle A = 31°, and we need to find the length of side CA (denoted as $x$). 2. **

1. **State the problem:** Find the maximum value of the function on the interval $[0, \frac{\pi}{2}]$. Since the function is not specified, we assume a common trigonometric functio

1. **Problem:** Find the value of $\tan \left[2 \cos \left(2 \sin^{-1} \frac{1}{2}\right)\right]$. 2. **Step 1: Evaluate the inner inverse sine function.**

1. **Problem:** Given $\sin \theta = \frac{3}{5}$ and $\theta$ is in quadrant I, find $\cos 2\theta$. 2. **Formula:** Use the double-angle identity for cosine:

1. The problem states that $\sin A = \frac{13}{12}$. We need to analyze this value. 2. Recall that the sine of an angle in a right triangle or on the unit circle must satisfy $-1 \

1. **State the problem:** We have two right triangles and need to find the primary trigonometric ratios (sine, cosine, tangent) for angle $A$ in each.

1. **State the problem:** An aeroplane leaves airport A and flies on a bearing of 035° for 1.5 hours at 600 km/h to airport B. Then it flies on a bearing of 130° for 1.5 hours at 4

1. **State the problem:** An aeroplane flies due north from Ikeja airport for 500 km, then flies on a bearing of 060° for 300 km, and finally flies over a road junction. We need to

1. **State the problem:** Prove that $$1 + \tan A + \sec A = \frac{2}{1 + \cot A - \csc A}$$. 2. **Recall the definitions:**

1. **Énoncé du problème :** On a un triangle ABC avec AB = $\sqrt{3}$, AC = 2, et BC = 1. Il faut montrer que ABC est un triangle rectangle.

1. **Problem Statement:** Given angles in degrees, we need to:

1. **State the problem:** Solve for $x$ in the equation $$4 \sin^2 x - 4 \sin x + 1 = 0$$ where $$0^\circ \leq x \leq 360^\circ$$. 2. **Identify the substitution:** Let $$y = \sin

1. **State the problem:** Prove the identity $$\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = 2 \sin 2x$$. 2. **Recall formulas and rules:**

1. The problem is to simplify the expression $\cos 80^\circ \cos 20^\circ + \sin 80^\circ \sin 20^\circ$.\n\n2. We use the cosine addition formula: $$\cos(A - B) = \cos A \cos B +