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Home Question Bank Math Trigonometry Archive 2025 January 01

Trigonometry Questions from Jan 01,2025

Browse the Trigonometry Q&A Archive for Jan 01,2025, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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8. ¿A qué distancia de la base de un faro de 6 m se encuentra un barco cuando este se observa desde la parte superior del faro con un ángulo de depresión \( x \) tal que sen \( x-\sqrt{3} \cos x=0 \) ? \( \begin{array}{llll}\text { A) } \sqrt{3} & \text { B) } 3 \sqrt{3} & \text { C) } 6 \sqrt{3} & \text { D) } 2 \sqrt{3}\end{array} \) \( \left\{\begin{array}{ll}7 & \text { Simplifier : } \\ \boldsymbol{a}= & \cos 40^{\circ}-3 \sin 70^{\circ}-\sin 50^{\circ}+3 \cos 20^{\circ} \\ b= & \sqrt{3} \cos ^{2} 36^{\circ}-\sqrt{3} \tan 66^{\circ} \times \tan 24^{\circ}+\frac{\sqrt{12}}{2} \cos ^{2} 54^{\circ}\end{array}\right. \) A right triangle has an angle measuring 60 degrees and its hypotenuse measures 10 cm. Find the measure of the side opposite to the 60-degree angle. Solving Trig Equations Solve the following trig equations. \( \begin{array}{ll}\text { 48. } \frac{1}{2} \cos x=-\frac{\sqrt{2}}{4} & \text { 49. }-5-\frac{1}{5} \tan x=-5\end{array} \) Proving Trig Identities Prove the following using trigonometric identities. \( \begin{array}{ll}\text { 44. } \frac{\tan ^{2} x}{\sin ^{2} x}=\frac{1}{\cos ^{2} x} & \text { 45. } \sec ^{2} x+\csc ^{2} x=\frac{\csc ^{2} x}{\cos ^{2} x}\end{array} \) \( \begin{array}{ll}\text { 46. } \cot ^{2} x-\cot ^{2} x \cos ^{2} x=\sec ^{2} x & \text { 47. } 2-\frac{\cos ^{2} x}{1-\sin ^{2} x}=1\end{array} \) Proving Trig Identities Prove the following using trigonometric identities. \( \begin{array}{lll}\text { 44. } \frac{\tan ^{2} x}{\sin ^{2} x}=\frac{1}{\cos ^{2} x} & \text { 45. } \sec ^{2} x+\csc ^{2} x=\frac{\csc ^{2} x}{\cos ^{2} x}\end{array} \) 3. La distancia de una persona a la base de un edificio es de 13 m . En un momento, una persona observa la parte más alta con un ángulo de elevación de \( 30^{\circ} \), después, la persona dice que camina hacía el edificio, tal que ahora ve la parte alta del edificio con un ángulo de elevación \( x \) tal que \( \cos \left(30^{\circ}+\boldsymbol{x}\right) \) - sen \( x=0 \). ¿Cuántos metros se movió la persona? \( \begin{array}{llll}\text { Al } \frac{13}{\sqrt{3}} \text { metro } & \text { B) } 1 \text { metro } & \text { C) } 0 \text { metros } & \text { D) } 13 \sqrt{3} \text { metros }\end{array} \) Simplifying Trig Identities Simplify the following using trig identities. 40. \( \tan ^{2} x \csc ^{2} x \) 41. \( \sec x-\sin x \tan x \) 42. \( \cos x(1+\sec x)(1-\cos x) \) 43. \( \sin ^{2} x+\cot ^{2} x \sin ^{2} x \) Simplifying Trig Identities Simplify the following using trig identities. \( \begin{array}{ll}\text { 40. } \tan ^{2} x \csc ^{2} x & \text { 41. } \sec x-\sin x \tan x\end{array} \) What is the definition of the cosine function in relation to a right triangle using the adjacent side and hypotenuse?
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