1st Year Math Notes Chapter 12
Handwritten notes of 1st Year Math Notes Chapter 12 written by Professor M. Sulman Sherazi Suib. These notes are very helpful in the preparation of Application of Trigonometry for the students of Mathematics of the Intermediate and these are according to the paper patterns of all Punjab boards.
Summary and Contents:
Topics which are discussed in the notes are given below:
- Solution of the triangle tables of trigonometric ratios
- Solution of right Triangles with its examples
- Our comprehensive 1st Year Math Notes Chapter 12 will ensure you're fully prepared for your exams.
- What is the angle of elevation with its diagram angle of depression with its diagram and Heights and distance angles of elevation and depression?
- Oblique Triangles with examples
- The law of cosines with its proofs
- The law of Tangents with its proofs
- Circle is connected to a triangle
- Circumcircles with diagram In-circle with its diagram
- Escribed Circles with diagram
- A triangle has six important elements; three angles and three sides. In a triangle ABC, the measures of the three angles are usually denoted by a, b, g and the measures of the three sides opposite to them are denoted by a, b, c respectively. If any three out of these six elements, out of which atleast one side, are given, the remaining three elements can be determined This process of inding the unknown elements is called the solution of the triangle. We have calculated the values of the trigonometric functions of the angles measuring 0°, 30°, 45°, 60° and 90°. But in a triangle, the angles are not necessarily of these few measures. So, in the solution of triangles, we may have to solve problems involving angles of measures other than these. In such cases, we shall have to consult natural sin/cos/tan tables or we may use sin , cos , tan keys on the calculator. Tables/calculator will also be used for inding the measures of the angles when value of trigonometric ratios are given e.g. to ind q when sinq = x.
- Tables of Trigonometric Ratios: Mathematicians have constructed tables giving the values of the trigonometric ratios of large number of angles between 0° and 90°. These are called tables of natural sines, cosines, tangents etc. In four-igure tables, the interval is 6 minutes and diference corresponding to 1,2, 3, 4, 5 minutes are given in the diference columns. The following examples will illustrate how to consult these tables.
- Example 1: Find the value of i) sin 38° 24’ ii) sin 38° 28’ iii) tan 65° 30’. Solution: In the irst column on the left hand side headed by degrees (in the Natural Sine table) we read the number 38°. Looking along the row of 38° till the minute column number 24’ is reached, we get the number 0.6211.
- To ind sin 38° 28’, we irst ind sin 38° 24’, and then see the right hand column headed by mean diferences. Running down the column under 4’ till the row of 38° is reached. We ind 9 as the diference for 4’. Adding 9 to 6211, we get 6220.
- Note: 1. As sin q, sec q and tan q go on increasing as q increases from 0° to 90°, so the numbers in the columns of the diferences for sin q, sec q and tan q are added. 2. Since cos q, cosec q and cot q decrease as q increases from 0° to 90°, therefore, for cos q, cosec q and cot q the numbers in the column of the, diferences are subtracted.
- Turning to the tables of Natural Tangents read the number 65° in the irst column on the left hand side headed by degrees. Looking along the row of 65° till the minute column under 30’ is reached, we get the number 1943. The integral part of the igure just next to 65° in the horizontal line is 2.