📏 trigonometry

1. **Problem Statement:** Find the expansions of $\sin(A+B+C)$, $\cos(A+B+C)$, and $\tan(A+B+C)$.\n\n2. **Expansion of $\sin(A+B+C)$:** Use the angle addition formula for sine twic

1. **Problem statement:** Find the expansions of $\sin(A + B + C)$, $\cos(A + B + C)$, and $\tan(A + B + C)$. 2. **Expand $\sin(A + B + C)$:** Use the angle sum formula for sine:

1. **State the problem:** Solve the equation $\tan x = \cot x + 2 \tan(2x - 90^\circ)$. 2. **Rewrite trigonometric functions:** Recall that $\cot x = \frac{1}{\tan x}$ and $\tan(2x

1. The problem is to calculate $\cos 75^\circ$ and round the result to 2 decimal places. 2. We can use the angle sum identity for cosine: $$\cos(a+b) = \cos a \cos b - \sin a \sin

1. **State the problem:** (i) Prove the identity $$\frac{\cos\theta}{\tan\theta(1-\sin\theta)} \equiv 1 + \sin\theta.$$

1. **State the problem:** Prove the identity \( \frac{\cos \theta}{\tan \theta} (1 - \sin \theta) \equiv 1 + \frac{1}{\sin \theta} \). Then solve \( \frac{\cos \theta}{\tan \theta}

1. The problem is to analyze the function $y = \sin 5x \cdot \cos 3x$. 2. We can use the product-to-sum identity for sine and cosine:

1. **State the problem:** We need to show that the equation $$2 \tan^2 \theta \sin^2 \theta = 1$$ can be rewritten as $$2 \sin^4 \theta + \sin^2 \theta - 1 = 0$$. 2. **Recall the i

1. সমীকরণ: $\sec 5x = \csc 10x$ সমাধান করব। 2. $\sec 5x = \frac{1}{\cos 5x}$ এবং $\csc 10x = \frac{1}{\sin 10x}$ তাই সমীকরণ হবে:

1. The problem states that we want to find $h$ given the equation $\cos(60) = \frac{1}{h}$. 2. Recall that $\cos(60^\circ) = \frac{1}{2}$.

1. **State the problem:** We have points O, A, and B with distances OA = 20 km, OB = unknown, and AB = 50 km. Bearings from O to A is 334° and from O to B is 54°. We want to find t

1. **State the problem:** We have two yachts starting from point O. Yacht A sails 20 km at a bearing of 334° from O. Yacht B sails such that point B is 50 km from A, and the bearin

1. **State the problem:** We have points O, A, and B with distances OA = 20 km and OB unknown, but AB = 50 km. Bearings from O are 334° to A and 54° to B. We want the bearing of A

1. **State the problem:** Given the equation $7\cos\theta - \sin\theta = 5$ and the condition $\tan\theta > 0$, find $\tan\theta$. 2. **Rewrite the equation:** We want to express t

1. **State the problem:** Given the equation $7\cos\theta - \sin\theta = 5$ and the condition $\tan\theta > 0$, find $\tan\theta$. 2. **Rewrite the equation:** We have

1. **Stating the problem:** Given that $A + B = \frac{\pi}{4}$, we want to simplify the expression $\frac{\cos B - \sin B}{\cos B + \sin B}$. 2. **Rewrite the expression:** The exp

1. The problem is to find an expression for $\tan 4\theta$ in terms of $\tan \theta$. 2. Use the double-angle formula for tangent:

1. **State the problem:** Solve the equation $$\sec 2\theta - 2 \tan 2\theta + 2 \tan \theta = 0$$ for $$0 < \theta < 360$$ degrees. 2. **Rewrite in terms of sine and cosine:** Rec

1. **State the problem:** Prove that $$\sec^2\theta - 2\tan^2\theta + 2\tan\theta = 0$$ and find the values of $$\theta$$. 2. **Recall identities:** We know that $$\sec^2\theta = 1

1. The problem is to solve the equation $$0 = -4\sin(x) - 4\cos(2x)$$ for $x$. 2. Start by rewriting the equation:

1. **State the problem:** We are given a triangle with points A, B, and C. We know |AB| = 8 km, |BC| = 13 km, and the bearing of B from C is 230°. We need to find: (a) the distance