Handwritten notes of 1st Year Math Notes Chapter 10 written by Professor M. Sulman Sherazi Suib. These notes are very helpful in the preparation of Trigonometric Identities Class 11 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:
- Our comprehensive 1st Year Math Notes Chapter 10 will ensure you're fully prepared for your exams.
- What is the distance formula with examples
- Fundamental law of trigonometry
- Remember some basic results of allied angles
- Trigonometric ratios of allied angles
- Double angle identities half angle identities triple angle identities
- Sum, difference, and products of sine and cosine
- In this section, we shall irst establish the fundamental law of trigonometry before discussing the Trigonometric Identities. For this we should know the formula to ind the distance between two points in a plane. i.e., square root o f the sum of square the diference of x-coordinates and square the diference of y-coordinates.
- Trigonometric Ratios of Allied Angles: The angles associated with basic angles of measure q to a right angle or its multiples are called allied angles. So, the angles of measure 90° ± θ , 180° ± θ , 270° ± θ , 360° ± θ , are known as allied angles. Using fundamental law, cos( - ) = cos cos + sin sin ab a b a b and its deductions, we derive the following identities:
- Note: The above results also apply to the reciprocals of sine, cosine and tangent. These results are to be applied frequently in the study of trigonometry, and they can be remembered by using the following device:
- If q is added to or subtracted from odd multiple of right angle, the trigonometric ratios change into co-ratios and vice versa.
- If q is added to or subtracted from an even multiple of 2p the trigonometric ratios shall remain the same.
- So far as the sign of the results is concerned, it is determined by the quadrant in which the terminal arm of the angle lies.
- In cosθ, cosθ, cos(2θ) and cos(2θ) even multiples of pi / 2 are involved. ∴ cos will remain as cosθ.
- Moreover, the angle of measure
- i) (pi - θ) will have terminal side in Quad. II, ∴ cos(pi - θ ) = - cosθ
- ii) (pi + θ) will have terminal side in Quad. III, ∴ cos(pi + θ ) = cosθ
- iii) (2pi - θ) will have terminal side in Quad. IV: ∴ cos(2pi - θ ) = cosθ
- iv) (2pi + θ) will have terminal side in Quad. I ∴ cos(2pi + θ ) = cosθ
- Without using the tables, ind the values of:
- i) sin ( -780°) ii) cot ( -855°) iii) csc(-2040°) iv) sec( -960°) v) tan (1110°) vi) sin ( -300°)
- Express each of the following as a trigonometric function of an angle of positive degree measure of less than 45°.
- i) sin 196° ii) cos 147° iii) sin 319° iv) cos 254° v) tan 294° vi) cos 728° vii) sin ( -625° ) viii) cos( -435°) ix) sin 150°
- Reduce sin4θ to an expression involving only function of multiples of q , raised to the irst power.
- Find the values of sin and cosθ θ without using table or calculator, when θ is i) 18° ii) 36° iii) 54° iv) 72°.
- Reduce cos4θ to an expression involving only function of multiples of θ , raised to the irst power.
- Half angle Identities: The formulas proved above can also be written in the form of half angle identities, in the following way:
- Express 3sin (θ+ 4) cosθ in the form rsin(θ + φ), where the terminal side of the angle of measure φ is in the I quadrant.