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What is Trigonometry? Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. The word trigonometry is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides’ ) and ‘metron’ (meaning measure). Trigonometric ratios of an angle are some ratios of the sides of a right triangle with respect to its acute angles. Trigonometric identities are some trigonometric ratios for some specific angles and some identities involving these ratios. Trigonometry in the modern sense began with the Greeks. Hipparchus(c. 190–120 BC) was the first to construct a table of values for a trigonometric function. He considered every triangle—planar or spherical—as being inscribed in a circle, so that each side becomes a chord. To compute the various parts of the triangle, one has to find the length of each chord as a function of the central angle that subtends it—or, equivalently, the length of a chord as a function of the corresponding arc width. This became the chief task of trigonometry for the next several centuries. As an astronomer, Hipparchus was mainly interested in spherical triangles, such as the imaginary triangle formed by three stars on the celestial sphere, but he was also familiar with the basic formulas of plane trigonometry. In Hipparchus’s time these formulas were expressed in purely geometric terms as relations between the various chords and the angles (or arcs) that subtend them; the modern symbols for the trigonometric functions were not introduced until the 17th century. Example Suppose the students of a school are visiting Eiffel tower . Now, if a student is looking at the top of the tower, a right triangle can be imagined to be made as shown in figure. Can the student find out the height of the tower, without actually measuring it? Yes the student can find the height of the tower with the help of trigonometry Line of Sight Horizontal SHIVANI SLAUJA Angle of Elevation The angle which the line of sight makes with a horizontal line drawn away from their eyes is called the angle of Elevation of aero plane from them. Angel of Elevation SHIVANI SLAUJA Angle of Depression The angle below horizontal that an observer must look to see an object that is lower than the observer. Note: The angle of depression is congruent to the angle of elevation (this assumes the object is close enough to the observer so that the horizontals for the observer and the object are effectively parallel). Horizontal Angel of Depression Trigonometric Ratios TYPES Student’S work Angle of Elevation • A man who is 2 m tall stands on horizontal ground 30 m from a tree. The angle of elevation of the top of the tree from his eyes is 28˚. Estimate the height of the tree. Solution: Let the height of the tree be h. Sketch a diagram to represent the situation. tan 28˚ = h – 2 = 30 tan 28˚ h = (30 ´ 0.5317) + 2 ← tan 28˚ = 0.5317 = 17.951 The height of the tree is approximately 17.95 m. Tarunikka singh Angle of Depression • The angle of depression of a vehicle on the ground from the top of tower is 60. If the vehicle is at distance of 100 meters away from the building, find the height of the tower. Solution: PQ is the height of the tower and RQ is the distance between the tower and the vehicle whereas PS is the line of sight. Angle of depression, ∠∠ SPR 60 degree. tanθ= Opposite side\Adjacent side Tan 60° = PQ\RQ h = 100 * tan 60° h = 173.20 The Height of the tower from ground is 173.20 meter. Amisha aggarwal Two Angle of Elevation Soumya narula Two Angle of Depression Question: From the top of a tower 100 m high,a man observes two cars on the opposite sides of the tower with angles of depression 30 (degree) and 45 (degree)respectively. Find the distance between the cars. Aditi jauhari Ques- A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point. Answer Let AB be the tower. D is the initial and C is the final position of the car respectively. Angles of depression are measured from A. BC is the distance from the foot of the tower to the car. A/q, In right ΔABC, tan 60° = AB/BC ⇒ √3 = AB/BC ⇒ BC = AB/√3 m also, In right ΔABD, tan 30° = AB/BD ⇒ 1/√3 = AB/(BC + CD) ⇒ AB√3 = BC + CD ⇒ AB√3 = AB/√3 + CD ⇒ CD = AB√3 - AB/√3 ⇒ CD = AB(√3 - 1/√3) ⇒ CD = 2AB/√3 Here, distance of BC is half of CD. Thus, the time taken is also half. Time taken by car to travel distance CD = 6 sec. Time taken by car to travel BC = 6/2 = 3 sec.