#### Chapters

## Chapter 2: Trigonometry - 1

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 2 Trigonometry - 1 Exercise 2.1 [Pages 21 - 22]

Find the trigonometric function of :

0°

Find the trigonometric function of :

30°

Find the trigonometric function of :

45°

Find the trigonometric function of :

60°

Find the trigonometric function of :

150°

Find the trigonometric functions of :

180°

Find the trigonometric function of :

210°

Find the trigonometric function of :

300°

Find the trigonometric functions of :

330°

Find the trigonometric functions of :

−30°

Find the trigonometric functions of :

−45°

Find the trigonometric functions of :

−60°

Find the trigonometric functions of :

− 90°

Find the trigonometric functions of :

−120°

Find the trigonometric functions of :

−225°

Find the trigonometric functions of :

−240°

Find the trigonometric functions of :

−270°

Find the trigonometric functions of :

−315°

State the signs of tan 380°

State the signs of cot 230°

State the signs of sec 468°

State the signs of cos4^{c} and cos4°. Which of these two functions is greater?

State the quadrant in which θ lies if :

sin θ < 0 and tan θ > 0

State the quadrant in which θ lies if :

cos θ < 0 and tan θ > 0

Evaluate the following:

sin 30° + cos 45° + tan 180°

Evaluate the following :

cosec 45° + cot 45° + tan 0°

Evaluate the following :

sin 30° × cos 45° × tan 360°

Find all trigonometric functions of angle in standard position whose terminal arm passes through point (3, −4).

If cos θ = `12/13`, 0 < θ < `pi/2`, find the value of `(sin^2theta - cos^2theta)/(2sinthetacostheta), 1/(tan^2theta)`

Using tables evaluate the following :

4 cot 45° – sec^{2} 60° + sin 30°

Using tables evaluate the following :

`cos^2 0 + cos^2 pi/6 + cos^2 pi/3 + cos^2 pi/2`

Find the other trigonometric functions:

If cos θ = `-3/5` and 180° < θ < 270°.

Find the other trigonometric functions:

If sec A = `-25/7` and A lies in the second quadrant.

Find the other trigonometric functions:

If cot x = `3/4`, x lies in the third quadrant.

Find the other trigonometric functions:

If tan x = `(-5)/12`, x lies in the fourth quadrant.

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 2 Trigonometry - 1 Exercise 2.2 [Page 31]

If 2 sinA = 1 = `sqrt(2)` cosB and `pi/2` < A < `pi`, `(3pi)/2` < B < `2pi`, then find the value of `(tan"A" + tan"B")/(cos"A" - cos"B")`

If `sin"A"/3 = sin"B"/4 = 1/5` and A, B are angles in the second quadrant then prove that 4cosA + 3cosB = – 5.

If tanθ = `1/2`, evaluate `(2sin theta + 3cos theta)/(4cos theta + 3sin theta)`

Eliminate θ from the following :

x = 3secθ , y = 4tanθ

Eliminate θ from the following :

x = 6cosecθ , y = 8cotθ

Eliminate θ from the following :

x = 4cosθ − 5sinθ, y = 4sinθ + 5cosθ

Eliminate θ from the following :

x = 5 + 6cosecθ, y = 3 + 8cotθ

**Eliminate θ from the following:**

2x = 3 − 4tanθ, 3y = 5 + 3secθ

If 2sin^{2}θ + 3sinθ = 0, find the permissible values of cosθ.

If 2cos^{2}θ − 11cosθ + 5 = 0 then find possible values of cosθ.

Find the acute angle θ such that 2 cos^{2}θ = 3 sin θ.

Find the acute angle θ such that 5tan^{2}θ + 3 = 9secθ.

Find sinθ such that 3cosθ + 4sinθ = 4

If cosecθ + cotθ = 5, then evaluate secθ.

If cotθ = `3/4` and π < θ < `(3pi)/4` then find the value of 4cosecθ + 5cosθ.

Find the Cartesian co-ordinates of points whose polar coordinates are :

(3, 90°)

Find the Cartesian co-ordinates of points whose polar coordinates are :

(1,180°)

Find the polar coordinates of points whose cartesian coordinates are :

(5, 5)

Find the polar coordinates of points whose cartesian coordinates are :

`(1, sqrt(3))`

Find the polar co-ordinates of points whose Cartesian co-ordinates are:

(–1, –1)

Find the polar co-ordinates of points whose Cartesian co-ordinates are:

`(- sqrt(3), 1)`

Find the value of :

`sin (19pi^"c")/3`

Find the value of :

cos 1140°

Find the values of:

`cot (25pi^"c")/3`

Prove the following identities:

`(1 + tan^2 "A") + (1 + 1/tan^2"A") = 1/(sin^2 "A" - sin^4"A")`

Prove the following identities:

(cos^{2}A – 1) (cot^{2}A + 1) = −1

Prove the following identities:

(sinθ + sec θ)^{2} + (cosθ + cosec θ)^{2} = (1 + cosecθ sec θ)^{2}

Prove the following identities:

(1 + cot θ – cosec θ)(1 + tan θ + sec θ) = 2

Prove the following identities:

`tan^3theta/(1 + tan^2theta) + cot^3theta/(1 + cot^3theta` = secθ cosecθ – 2sinθ cosθ

Prove the following identities:

`1/(sectheta + tantheta) - 1/costheta = 1/costheta - 1/(sectheta - tantheta)`

Prove the following identities:

`sintheta/(1 + costheta) + (1 + costheta)/sintheta` = 2cosecθ

Prove the following identities:

`tantheta/(sectheta - 1) = (sectheta + 1)/tantheta`

Prove the following identities:

`cottheta/("cosec" theta - 1) = ("cosec" theta + 1)/cot theta`

Prove the following identities:

(secA + cosA)(secA − cosA) = tan^{2}A + sin^{2}A

Prove the following identities:

1 + 3cosec^{2}θ cot^{2}θ + cot^{6}θ = cosec^{6}θ

**Prove the following identities:**

`(1 - sec theta + tan theta)/(1 + sec theta - tan theta) = (sec theta + tan theta - 1)/(sec theta + tan theta + 1)`

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 2 Trigonometry - 1 Miscellaneous Exercise 2 [Pages 32 - 34]

Select the correct option from the given alternatives:

The value of the expression cos1°. cos2°. cos3° … cos179° =

– 1

0

`1/sqrt(2)`

1

Select the correct option from the given alternatives:

`tan"A"/(1 + sec"A") + (1 + sec"A")/tan"A"` is equal to

2cosecA

2secA

2sinA

2cosA

Select the correct option from the given alternatives:

If α is a root of 25cos^{2}θ + 5cosθ – 12 = 0, `pi/2` < α < π, then sin2α is equal to

`-24/25`

`-13/18`

`13/18`

`24/25`

Select the correct option from the given alternatives:

If θ = 60°, then `(1 + tan^2theta)/(2tantheta)` is equal to

`sqrt(3)/2`

`2/sqrt(3)`

`1/sqrt(3)`

`sqrt(3)`

Select the correct option from the given alternatives:

If secθ = m and tanθ = n, then `1/"m"{("m + n") + 1/(("m + n"))}` is equal to

2

mn

2m

2n

Select the correct option from the given alternatives:

If cosecθ + cotθ = `5/2`, then the value of tanθ is

`14/25`

`20/21`

`21/20`

`15/16`

Select the correct option from the given alternatives:

`1 - sin^2theta/(1 + costheta) + (1 + costheta)/sintheta - sintheta/(1 - costheta)` equals

0

1

sin θ

cos θ

Select the correct option from the given alternatives:

If cosecθ − cotθ = q, then the value of cot θ is

`(2"q")/(1 + "q"^2)`

`(2"q")/(1 - "q"^2)`

`(1 - "q"^2)/(2"q")`

`(1 + "q"^2)/(2"q")`

Select the correct option from the given alternatives:

The cotangent of the angles `pi/3, pi/4 and pi/6` are in

A.P.

G.P.

H.P.

Not in progression

Select the correct option from the given alternatives:

The value of tan1°.tan2°tan3°..... tan89° is equal to

– 1

1

`pi/2`

2

Answer the following:

Find the trigonometric functions of :

90°

Find the trigonometric functions of:

120°

Answer the following:

Find the trigonometric functions of :

225°

Answer the following:

Find the trigonometric functions of :

240°

Answer the following:

Find the trigonometric functions of :

270°

Answer the following:

Find the trigonometric functions of :

315°

Find the trigonometric functions of :

−120°

Answer the following:

Find the trigonometric functions of :

−150°

Answer the following:

Find the trigonometric functions of :

−180°

Answer the following:

Find the trigonometric functions of :

−210°

Answer the following:

Find the trigonometric functions of :

−300°

Answer the following:

Find the trigonometric functions of :

−330°

Answer the following:

State the signs of cosec 520°

Answer the following:

State the signs of cot 1899°

Answer the following:

State the signs of sin 986°

Answer the following:

State the quadrant in which θ lies if tan θ < 0 and sec θ > 0

Answer the following:

State the quadrant in which θ lies if sin θ < 0 and cos θ < 0

Answer the following:

State the quadrant in which θ lies if sin θ > 0 and tan θ < 0

Answer the following:

Which is greater sin(1856°) or sin(2006°)?

Answer the following:

Which of the following is positive? sin(−310°) or sin(310°)

Answer the following:

Show that 1 − 2sinθ cosθ ≥ 0 for all θ ∈ R.

Answer the following:

Show that tan^{2}θ + cot^{2}θ ≥ 2 for all θ ∈ R

Answer the following:

If sinθ = `(x^2 - y^2)/(x^2 + y^2)` then find the values of cosθ, tanθ in terms of x and y.

Answer the following:

If sec θ = `sqrt(2)` and `(3pi)/2 < theta < 2pi` then evaluate `(1 + tantheta + "cosec"theta)/(1 + cottheta - "cosec"theta)`

Prove the following:

sin^{2}A cos^{2}B + cos^{2}A sin^{2}B + cos^{2}A cos^{2}B + sin^{2}A sin^{2}B = 1

Prove the following:

`((1 + cot theta + tan theta)(sin theta - costheta)) /(sec^3theta - "cosec"^3theta)`= sin^{2}θ cos^{2}θ

Prove the following:

`(tan theta + 1/costheta)^2 + (tan theta - 1/costheta)^2 = 2((1 + sin^2theta)/(1 - sin^2theta))`

Prove the following:

2 sec^{2}θ – sec^{4}θ – 2cosec^{2}θ + cosec^{4}θ = cot^{4}θ – tan^{4}θ

Prove the following:

sin^{4}θ + cos^{4}θ = 1 – 2 sin^{2}θ cos^{2}θ

Prove the following:

2(sin^{6}θ + cos^{6}θ) – 3(sin^{4}θ + cos^{4}θ) + 1 = 0

Prove the following:

cos^{4}θ − sin^{4}θ +1= 2cos^{2}θ

Prove the following:

sin^{4}θ +2sin^{2}θ . cos^{2}θ = 1 − cos^{4}θ

Prove the following:

`(sin^3theta + cos^3theta)/(sintheta + costheta) + (sin^3theta - cos^3theta)/(sintheta - costheta)` = 2

Prove the following:

tan^{2}θ − sin^{2}θ = sin^{4}θ sec^{2}θ

Prove the following:

(sinθ + cosecθ)^{2} + (cosθ + secθ)^{2 }= tan^{2}θ + cot^{2}θ + 7

Prove the following:

sin^{8}θ − cos^{8}θ = (sin^{2}θ − cos^{2}θ) (1 − 2 sin^{2}θ cos^{2}θ)

Prove the following:

sin^{6}A + cos^{6}A = 1 − 3sin^{2}A + 3 sin^{4}A

Prove the following:

(1 + tanA · tanB)^{2} + (tanA − tanB)^{2} = sec^{2}A · sec^{2}B

Prove the following:

`(1 + cot + "cosec" theta)/(1 - cot + "cosec" theta) = ("cosec" theta + cottheta - 1)/(cottheta - "cosec"theta + 1)`

Prove the following:

`(tantheta + sectheta - 1)/(tantheta + sectheta + 1) = tantheta/(sec theta + 1)`

Prove the following:

`("cosec"theta + cottheta - 1)/( "cosec"theta + cot theta + 1) =(1-sintheta)/costheta`

Prove the following:

`("cosec"theta + cottheta + 1)/(cottheta + "cosec" theta - 1) = cottheta/("cosec"theta - 1)`

## Chapter 2: Trigonometry - 1

## Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 2 - Trigonometry - 1

Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 2 (Trigonometry - 1) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 2 Trigonometry - 1 are Introduction of Trigonometry, Trigonometric Functions with the Help of a Circle, Signs of Trigonometric Functions in Different Quadrants, Range of Cosθ and Sinθ, Trigonometric Functions of Specific Angles, Trigonometric Functions of Negative Angles, Fundamental Identities, Periodicity of Trigonometric Functions, Domain and Range of Trigonometric Functions, Graphs of Trigonometric Functions, Polar Co-ordinate System.

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