SOLUTION OF RIGHT TRIANGLE Solving a triangle means determining the measures of all sides and angles of the triangles. In solving problems involving right triangle, the following steps may be considered:
Sketch the required triangle as accurate as possible based on the given data.
Identify the given and the required parts of the triangle.
Solve for the unknown parts of the triangle using any of the following:
The definitions of the trigonometric functions. b) The Pythagorean relations. c) The relation of complimentary angles.
EXAMPLE 1. Solve the following triangles, in which C = 900: a = 35 , c = 92 b) A = 29030’ , b = 72.8 2. Compute for the length of AD. A B 8 cm. 480 C 72010’ D
EXAMPLE 3. Compute for the missing parts of the given composite triangle. B A 630 16cm. C 700 D 280 E
EXAMPLE 4. Solve for LM. L M 48 25030’ 52025’ N K
ANGLES OF ELEVATION AND DEPRESSION If an observer sights an object, the angle formed between a horizontal line and his line of sight is called the angle of elevationif the line of sight is above the horizontal and the angle of depression if the line of sight is below the horizontal. Object Line of sight Angle of Elevation Horizontal Line Observer Angle of Depression Line of sight Object
EXAMPLE If the angle of elevation of the top of the tower is 52031’. Find the height of the tower if the observer is 41.5 m. from its base. 2. Find the angle of elevation of the sun if the shadow of the pole 60 ft. tall and reaches 90 ft. from the pole. 3. From the top of a lighthouse 30 m. high, the angle of depression of a boat in the sea was 28045’. How far was the boat from the top and base of lighthouse?
EXAMPLE 4. From a window Carlo observes the lamp post. He noted that the angle of elevation of its top is 43020’ while the angle of depression of its base is 20015’. If the top of the lamp post is 30 m. away from the window and its base is 25 m., what is the height of the lamp post? 5. From where he stands 75 ft. away from the tree, a mountaineer 6 ft. tall, found that the angle of elevation of the top of the tree was 37025’. Find the total height of the tree.
DIRECTION OF ANGLES There are two ways in which the direction of an angle can be determined. They are bearing and course. Bearing is an acute angle measured from due north or due south. The North-South line is the basis of the acute angle measurement. Course is the angle measured clockwise from north to the line of travel.
EXAMPLE Course readings of 750, 1500 and 3150 are illustrated below with their corresponding bearing readings: N N 750 N 3150 1500 S S N450W S300E N750E S
EXAMPLE 1.Towers A and B are on east-west line 125 m. apart. Jojo, on the north of that line finds that the direction of towers A and B are N39030’E and N50030’W, respectively. How far is Jojo from each tower? 2. From a boat sailing due north at 16 km/hr, a wrecked ship A and an observation tower B are observed in a line due east. One hour later the wrecked ship and the tower have bearings S350E and S650E. Find the distance between the wrecked ship and the tower. 3. A car travels 125 km. on a course of 38015’ then make a turn on a course of 128015’ at a distance of 315 km. Find the distance from the starting point to the end of the trip.
EXERCISES 1. Pepe and Pilar are 105 m. apart are on the same horizontal ground. Pepe sees that a kite is directly over the head of Pilar and its angle of elevation is 67020’. How high is the kite? How far is the kite from where Pepe stands? 2. The angle of elevation of the top of the tower is found to be 420 at point A and 520 at point B, 28 ft. nearer the tower. What is the height of the tower if both observation points and the base of the tower are in the same horizontal plane? 3. From a cliff 1500 m. high, the two lighthouses in opposite directions have angles of depression of 330 and 290, respectively. Find the distance between the two lighthouses.