All categories

3 years ago

Complete the following to make a true identity. 1. sin^2 3x + cos^2 3x = ?

Mathematics

2 answers:

Andru [333]3 years ago

8 0

Answer:

Step-by-step explanation:

sin^2+cos^2 A=1 is an indentity and is true for all A, including A= 3x and hence sin^2+cos^2 3x=1

However, let us try is using values of sin3x and cos3x, but before this as sin^2x + cos^2x =1, squaring sin^4 x + cos^4 x =2sin^2 x cos^2 x=1 or sin^4 x +cos^4 x= 1 - 2 sin^2 x and sin^6 x + cos^6 x=1 -3 sin^2 x cos^2 x (sin^2 x + cos^2 x) =1 -3 sin^2 x cos^2 x -as a^3 =b^3= (a+b)^3 -3ab(a+b)

Now coming to proof as

sin3x= 3 sin x- 4 sin^3 x and cos x= 4 cos^3 x - 3 sin x

Therefore

sin^2 3x +cos^2 3x=(3 sin x- 4 sin^3 x)^2 +(4cos^3 x - 3 cos x)^2

9sin^2 x = 16 sin^6 x -24 sin^4 x + 16 cos^6 x + 9 cos^2 x- 24 cos^4 x

9(sin^2 x +cos^2 x) + 16(sin^6 x + cos^6 x) - 24 (sin^4 x + cos^4 x)

9*1 +16(1-3 sin^2 x cos^2 x) - 24 (1-2 sin^2 x cos^2 x)

9+16-48sin^2 x cos^2 x -24+48 sin^2 x cos^2 x

andrey2020 [161]3 years ago

5 0

Answer:

Step-by-step explanation:

I have done this by a Calculator

Red

You might be interested in

Verify each trigonometric equation by substituting identities to match the right hand side of the equation to the left hand side

lorasvet [3.4K]

Answer:

Step-by-step explanation:

cot x sec⁴ x = cot x+2 tan x +tan³x

L.H.S = cot x sec⁴x

=cot x (sec²x)²

=cot x (1+tan²x)² [ ∵ sec²x=1+tan²x]

= cot x(1+ 2 tan²x +tan⁴x)

=cot x+ 2 cot x tan²x+cot x tan⁴x

=cot x +2 tan x + tan³x [ ∵cot x tan x $=\frac{ \textrm{tan x }}{\textrm{tan x}}$ =1]

=R.H.S

(sin x)(tan x cos x - cot x cos x)=1-2 cos²x

L.H.S =(sin x)(tan x cos x - cot x cos x)

= sin x tan x cos x - sin x cot x cos x

$=\textrm{sin x cos x }\times\frac{\textrm{sin x}}{\textrm{cos x} } - \textrm{sinx}\times\frac{\textrm{cos x}}{\textrm{sin x}}\times \textrm{cos x}$

= sin²x -cos²x

=1-cos²x-cos²x

=1-2 cos²x

=R.H.S

1+ sec²x sin²x =sec²x

L.H.S =1+ sec²x sin²x

= $1+\frac{{sin^2x}}{cos^2x}$ [ $\textrm{sec x}=\frac{1}{\textrm{cos x}}$ ]

=1+tan²x $[\frac{\textrm{sin x}}{\textrm{cos x}} = \textrm{tan x}]$

=sec²x

=R.H.S

$\frac{\textrm{sinx}}{\textrm{1-cos x}} +\frac{\textrm{sinx}}{\textrm{1+cos x}} = \textrm{2 csc x}$

L.H.S= $\frac{\textrm{sinx}}{\textrm{1-cos x}} +\frac{\textrm{sinx}}{\textrm{1+cos x}}$

$=\frac{\textrm{sinx(1+cos x)+{\textrm{sinx(1-cos x)}}}}{\textrm{(1-cos x)\textrm{(1+cos x})}}$

$=\frac{\textrm{sinx+sin xcos x+{\textrm{sinx-sin xcos x}}}}{{(1-cos ^2x)}}$

$=\frac{\textrm{2sin x}}{sin^2 x}$

= 2 csc x

= R.H.S

-tan²x + sec²x=1

L.H.S=-tan²x + sec²x

= sec²x-tan²x

= $\frac{1}{cos^2x} -\frac{sin^2x}{cos^2x}$

$=\frac{1- sin^2x}{cos^2x}$

$=\frac{cos^2x}{cos^2x}$

8 0

3 years ago

PART 3 OF ME ASKING FOR HELP

e-lub [12.9K]

Answer:

Step-by-step explanation:

7 0

2 years ago

PLEASE HELP! PLEASE HELP! PLEASE HELP

marin [14]

1Distributive
2Commutative
3Associative
4Commutative
5Associative

4 0

2 years ago

Toby wants to find the volume of a solid toy soldier.

mr Goodwill [35]

Answer:

Toby wants to find the volume of a solid toy soldier.He fills a rectangular container 8 cm long, 6 cm wide,and 10 cm high with water to a depth of 4 cm. Toby totally submerges the toy soldier in the water. The height of the water with the submerged toy soldier is 6.6 cm. Which of the following is closest to the volume, in cubic centimeters, of the toy soldier?

A. 125 B. 156 C. 192 D. 208 E.317

The volume of the toy is 125 cubic cm ,option A is the closest answer.

Step-by-step explanation:

Given:

A rectangular container .

Length of the container =8 cm

Width of the container = 6 cm

Height of the container when water is filled = $(h_1)$ = 4 cm