Question

GIVE ALL TRIGNOMETRY IDENTITIES AND RATIOS WITH EXAMPLES



MATHEMATHICS CLASS 10
CH.8 TRIGNOMETRY

{ 100 points }

••NEeD QUAliTy aNSwEr ••​

Answer 1

Answer:

Ello ❣️

Step-by-step explanation:

TRIGONOMETRIC RATIOS :-

sine = opposite side / hypotenuse

cosine = adjacent side / hypotenuse

tangent = opposite side / adjacent side

secant = hypotenuse / adjacent side

cosecant = hypotenuse / opposite side

cotangent = adjacent side / opposite side

TRIGONOMETRIC IDENTITIES :-

${sin}^{2} \alpha + {cos}^{2} \alpha = 1 \\ { \ \sec }^{2} \alpha \: - \: {tan}^{2} \alpha = 1 \\ {cosec}^{2} \alpha - {cot}^{2} \alpha = 1 \\ sin \alpha \times \frac{1}{cosec \alpha } = 1 \\ cos \alpha \times \frac{1}{sec \alpha } = 1 \\ tan \alpha \times \frac{1}{cot \alpha } = 1$

Answer 2

Answer :-

Now we have a triangle ABC right angled at B

We will consider the angle ACB .

▪️AC is the hypotenuse

▪️AB is Perpendicular to the Base BC

Let the side :-

▪️AB be 'p'

▪️AC be 'h'

▪️CB be 'b'

Now the ratio :-

$Sin\theta = \dfrac{\textsf{Side in front of angle}}{\textsf{ Hypotenuse}} = \dfrac{p}{h}$

$Cos\theta = \dfrac{\textsf{Side adjacent of angle}}{\textsf{ Hypotenuse}} = \dfrac{b}{h}$

$Tan\theta = \dfrac{\textsf{Side in front of angle}}{\textsf{Side adjacent of angle}} = \dfrac{p}{h}$

$Cot\theta = \dfrac{\textsf{Side Adjacent of angle}}{\textsf{ Side in front of angle}} = \dfrac{p}{h}$

$Sec\theta = \dfrac{\textsf{Hypotenuse}}{\textsf{Side adjacent of angle}} = \dfrac{p}{h}$

$Cosec\theta = \dfrac{\textsf{Hypotenuse}}{\textsf{ Side in front of angle}} = \dfrac{p}{h}$

Now Some basic Identities :-

▪️Sin²∅ + Cos²∅ = 1

▪️Sec²∅ - Tan²∅ = 1

▪️Cosec² - Cot²∅ = 1

$\bullet Tan\theta = \dfrac{Sin\theta}{Cos\theta}$

$\bullet Cot\theta = \dfrac{Cos\theta}{Sin\theta}$

$\bullet Cosec\theta + Cot\theta = \dfrac{1}{Cosec\theta - Cot\theta}$

$\bullet Sec\theta + Tan\theta = \dfrac{1}{Sec\theta - Tan\theta}$

Some results :-

▪️Sin(A + B) = Sin(A).Cos(B) + Sin(B).Cos(A)

▪️Sin(A - B) = Sin(A).Cos(B) - Sin(B).Cos(A)

▪️Cos(A + B) = Cos(A).Cos(B) - Sin(A).Sin(A)

▪️Cos(A - B) = Cos(A).Cos(B) + Sin(A).Sin(A)

$\bullet Tan(A+B)= \dfrac{Tan(A) + Tan(B)}{1 - Tan(A).Tan(B)}$

$\bullet Tan(A-B)= \dfrac{Tan(A) - Tan(B)}{1 + Tan(A).Tan(B)}$

▪️Sin(2A) = 2.Sin(A).Cos(A)

▪️Cos(2A) = Cos²(A) - Sin²(A)

$\bullet Tan(2A)= \dfrac{2Tan(A)}{1 - Tan^2(A)}$