Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

1,816 views

Published on

No Downloads

Total views

1,816

On SlideShare

0

From Embeds

0

Number of Embeds

6

Shares

0

Downloads

216

Comments

0

Likes

7

No embeds

No notes for slide

- 1. INTRODUCTION TO TRIGONOMETRY Submitted By AMAL A S
- 2. Trigonometry...????? The word trigonometry is derived from the ancient Greek language and means measurement of triangles. trigonon “triangle” + metron “measure” = Trigonometry
- 3. Trigonometry ... is all about Triangles… C a b B A c
- 4. A right-angled triangle (the right angle is shown by the little box in the corner) has names for each side: Adjacent is adjacent to the angle "θ“ Opposite is opposite the angle The longest side is the Hypotenuse. Opposite Right Angled Triangle θ Adjacent
- 5. DEGREE MEASURE AND RADIAN MEASURE B B 1 1 1 A O 1 A O Initial Side Degree measure: If a rotation from the initial side to terminal side is(1/360)th of a revolution, the angle is said to have a measure of one degree, written as 1 . Radian measure: Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have measure of 1 radian. Degree measure= 180/ π x Radian measure Radian measure= π/180 x Degree measure
- 6. ANGLES Angles (such as the angle "θ" ) can be in Degrees or Radians. Here are some examples: Angle Degree Radians Right Angle 90° π/2 Straight Angle 180° π Full Rotation 360° 2π
- 7. Trigonometric functions..
- 8. "Sine, Cosine and Tangent" The three most common functions in trigonometry are Sine, Cosine and Tangent. They are simply one side of a triangle divided by another. Sine Function: sin(θ) = Opposite / Hypotenuse Cosine Function: cos(θ) = Adjacent / Hypotenuse Tangent Function: tan(θ) = Opposite / Adjacent Opposite For any angle "θ": θ Adjacent
- 9. Their graphs as functions and the characteristics •
- 10. TRGONOMETRIC FUNCTIONS π/6 π/4 π/3 π/2 π 3π/2 2π 30° 0° 45° 60° 90° 180° 270° 360° sin 0 1/2 1/√2 √3/2 1 0 -1 0 cos 1 √3/2 1/√2 1/2 0 -1 0 1 tan 0 1/√3 1 √3 Not defined 0 Not defined 0
- 11. Other Functions (Cotangent, Secant, Cosecant) Cosecant Function : csc(θ) = Hypotenuse / Opposite Secant Function : sec(θ) = Hypotenuse / Adjacent Cotangent Function : cot(θ) = Adjacent / Opposite Opposite Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another: θ Adjacent
- 12. Proof for trigonometric ratios 30 ,45 ,60
- 13. Computing unknown sides or angles in a right triangle. In order to find a side of a right triangle you can use the Pythagorean Theorem, which is a2+b2=c2. The a and b represent the two shorter sides and the c represents the longest side which is the hypotenuse. To get the angle of a right angle you can use sine, cosine, and tangent inverse. They are expressed as tan^(-1) ,cos^(-1) , and sin^(1) .
- 14. o Find the sine, the cosine, and the tangent of 30 . Begin by sketching a 30 -60 -90 triangle. To make the calculations simple, you can choose 1 as the length of the shorter leg. From Pythagoras Theorem , it follows that the length of the longer leg is √3 and the length of the hypotenuse is 2. sin 30 = opp./hyp. = 1/2 = 0.5 cos 30 = adj./hyp. = √3/2 ≈ 0.8660 2 1 30 tan 30 = opp./adj. = 1/√3 = √3/3 ≈ 0.5774 √3
- 15. o Find the sine, the cosine, and the tangent of 45 . Begin by sketching a 45 -45 -90 triangle. Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg. The length of the hypotenuse is √2 (Pythagoras Theorem). sin 45 = opp./hyp. = 1/√2 =2/√2≈ 0.7071 cos 45 = adj./hyp. = 1/√2 =2/√2≈ 0.7071 √2 1 45 tan 45 = opp./adj. = 1/1 = 1 1
- 16. o Find the sine, the cosine, and the tangent of 60 . Begin by sketching a 30 -60 -90 triangle. To make the calculations simple, you can choose 1 as the length of the shorter leg. From Pythagoras Theorem , it follows that the length of the longer leg is √3 and the length of the hypotenuse is 2. sin 60 = opp./hyp = √3/2 ≈ 0.8660 60 2 1 cos 60 = adj./hyp = ½ = 0.5 tan 60 = opp./adj. = √3/1 ≈ 1.7320 30 √3
- 17. TRIGONOMETRIC IDENTITIES
- 18. Reciprocal Identities Pythagorean Identities sin u = 1/csc u sin2 u + cos2 u = 1 cos u = 1/sec u 1 + tan2 u = sec2 u tan u = 1/cot u 1 + cot2 u = csc2 u csc u = 1/sin u Quotient Identities sec u = 1/cos u tan u = sin u /cos u cot u = 1/tan u cot u =cos u /sin u
- 19. Co-Function Identities Parity Identities (Even & Odd) sin( π/2− u) = cos u sin(−u) = −sin u cos( π/2− u) = sin u cos(−u) = cos u tan( π/2− u) = cot u tan(−u) = −tan u cot( π/2− u) = tan u cot(−u) = −cot u csc( π/2− u) = sec u csc(−u) = −csc u sec( π/2− u) = csc u sec(−u) = sec u
- 20. Sum & Diﬀerence Formulas sin(u ± v) = sin u cos v ± cos u sin v cos(u ± v) = cos u cos v ∓ sin u sin v tan(u ± v) = tan u ± tan v / 1 ∓ tan u tan v Double Angle Formulas sin(2u) = 2sin u cos u cos(2u) = cos2 u − sin2 u = 2cos2 u − 1 = 1 − 2sin2 u tan(2u) =2tanu /(1 − tan2 u)
- 21. Sum-to-Product Formulas Sin u + sin v = 2sin [ (u + v) /2 ] cos [ (u − v ) /2 ] Sin u − sin v = 2cos [ (u + v) /2 ] sin [ (u − v ) /2 ] Cos u + cos v = 2cos [ (u + v) /2 ] cos [ (u − v) /2 ] Cos u − cos v = −2sin [ (u + v) /2 ] sin [ (u − v) /2 ]
- 22. Product-to-Sum Formulas Sin u sin v = ½ [cos(u − v) − cos(u + v)] Cos u cos v = ½ [cos(u − v) + cos(u + v)] Sin u cos v = ½ [sin(u + v) + sin(u − v)] Cos u sin v = ½ [sin(u + v) − sin(u − v)]
- 23. Power – Reducing / Half Angle Formulas sin2 u = 1 − cos(2u) / 2 cos2 u = 1 + cos(2u) / 2 tan2 u = 1 − cos(2u) / 1 + cos(2u)

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment