NCERT Exemplar solutions for Mathematics [English] Class 11 chapter 3 - Trigonometric Functions [Latest edition]

A circular wire of radius 3 cm is cut and bent so as to lie along the circumference of a hoop whose radius is 48 cm. Find the angle in degrees which is subtended at the centre of hoop.

If A = cos²θ + sin⁴θ for all values of θ, then prove that `3/4` ≤ A ≤ 1.

Find the value of `sqrt(3)` cosec 20° – sec 20°

If θ lies in the second quadrant, then show that `sqrt((1 - sin theta)/(1 + sin theta)) + sqrt((1 + sin theta)/(1 - sin theta))` = −2sec θ

Find the value of tan 9° – tan 27° – tan 63° + tan 81°

Prove that `(sec8 theta - 1)/(sec4 theta - 1) = (tan8 theta)/(tan2 theta)`

Solve the equation sin θ + sin 3θ + sin 5θ = 0

Solve 2 tan²x + sec²x = 2 for 0 ≤ x ≤ 2π.

Find the value of `(1 + cos  pi/8)(1 + cos  (3pi)/8)(1 + cos  (5pi)/8)(1 + cos  (7pi)/8)`

If x cos θ = `y cos (theta + (2pi)/3) = z cos (theta + (4pi)/3)`, then find the value of xy + yz + zx.

If α and β are the solutions of the equation a tan θ + b sec θ = c, then show that tan (α + β) = `(2ac)/(a^2 - c^2)`.

Show that 2 sin²β + 4 cos (α + β) sin α sin β + cos 2(α + β) = cos 2α

If angle θ is divided into two parts such that the tangent of one part is k times the tangent of other, and Φ is their difference, then show that sin θ = `(k + 1)/(k - 1)` sin Φ

Solve `sqrt(3)` cos θ + sin θ = `sqrt(2)`

If tan θ = `(-4)/3`, then sin θ is ______.

If sin θ and cos θ are the roots of the equation ax² – bx + c = 0, then a, b and c satisfy the relation ______.

The greatest value of sin x cos x is ______.

The value of sin 20° sin 40° sin 60° sin 80° is ______.

The value of `cos  pi/5 cos  (2pi)/5 cos  (4pi)/5 cos  (8pi)/5`  is ______.

If 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = ______.

“The inequality `2^sintheta + 2^costheta ≥ 2^(1/sqrt(2))` holds for all real values of θ” 

Match each item given under column C₁ to its correct answer given under column C₂.

Prove that `(tanA + secA  - 1)/(tanA - secA + 1) = (1 + sinA)/cosA`

If `(2sinalpha)/(1 + cosalpha + sinalpha)` = y, then prove that `(1 - cosalpha + sinalpha)/(1 + sinalpha)` is also equal to y.
`["Hint": "Express" (1 - cosalpha + sinalpha)/(1 + sinalpha) = (1 - cosalpha + sinalpha)/(1 + sinalpha) * (1 + cosalpha + sinalpha)/(1 + cosalpha + sinalpha)]`

If m sinθ = n sin(θ + 2α), then prove that tan(θ + α)cotα = `(m + n)/(m - n)`

[Hint: Express `(sin(theta + 2alpha))/sintheta = m/n` and apply componendo and dividendo]

If cos(α + β) = `4/5` and sin(α – β) = `5/13`, where α lie between 0 and `pi/4`, find the value of tan2α.
[Hint: Express tan2α as tan(α + β + α – β)]

If tanx = `b/a`, then find the value of `sqrt((a + b)/(a - b)) + sqrt((a - b)/(a + b))`

Prove that cosθ `cos  theta/2 - cos 3theta cos  (9theta)/2` = sin 7θ sin 8θ.

[Hint: Express L.H.S. = `1/2[2costheta cos  theta/2 - 2 cos 3theta cos  (9theta)/2]`

If a cosθ + b sinθ = m and a sinθ - b cosθ = n, then show that a² + b² = m² + n² 

Find the value of tan22°30′. `["Hint:"  "Let" θ = 45°, "use" tan  theta/2 = (sin  theta/2)/(cos  theta/2) = (2sin  theta/2 cos  theta/2)/(2cos^2  theta/2) = sintheta/(1 + costheta)]`

Prove that sin 4A = 4sinA cos³A – 4 cosA sin³A

If tanθ + sinθ = m and tanθ – sinθ = n, then prove that m² – n² = 4sinθ tanθ 
[Hint: m + n = 2tanθ, m – n = 2sinθ, then use m² – n² = (m + n)(m – n)]

If tan(A + B) = p, tan(A – B) = q, then show that tan 2A = `(p + q)/(1 - pq)`

If cosα + cosβ = 0 = sinα + sinβ, then prove that cos2α + cos2β = -2cos(α + β).
[Hint: (cosα + cosβ)² - (sinα + sinβ)² = 0]

If `(sin(x + y))/(sin(x - y)) = (a + b)/(a - b)`, then show that `tanx/tany = a/b` [Hint: Use Componendo and Dividendo].

If tanθ = `(sinalpha - cosalpha)/(sinalpha + cosalpha)`, then show that sinα + cosα = `sqrt(2)` cosθ.

[Hint: Express tanθ = `tan (alpha - pi/4) theta = alpha - pi/4`]

If sinθ + cosθ = 1, then find the general value of θ.

Find the most general value of θ satisfying the equation tan θ = –1 and cos θ = `1/sqrt(2)`.

If cotθ + tanθ = 2cosecθ, then find the general value of θ.

If 2sin²θ = 3cosθ, where 0 ≤ θ ≤ 2π, then find the value of θ.

If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.

If sin(θ + α) = a and sin(θ + β) = b, then prove that cos 2(α - β) - 4ab cos(α - β) = 1 - 2a² - 2b²

[Hint: Express cos(α - β) = cos((θ + α) - (θ + β))]

If cos(θ + Φ) = m cos(θ – Φ), then prove that 1 tan θ = `(1 - m)/(1 + m) cot phi`

[Hint: Express `(cos(theta + Φ))/(cos(theta - Φ)) = m/1` and apply Componendo and Dividendo]

Find the value of the expression `3[sin^4 ((3pi)/2 - alpha) + sin^4 (3pi + alpha)] - 2[sin^6 (pi/2 + alpha) + sin^6 (5pi - alpha)]`

If acos2θ + bsin2θ = c has α and β as its roots, then prove that tanα + tanβ = `(2b)/(a + c)`.

`["Hint: Use the identities" cos2theta = (1 - tan^2theta)/(1 + tan^2theta) "and" sin2theta =  (2tantheta)/(1 + tan^2theta)]`.

If x = sec Φ – tan Φ and y = cosec Φ + cot Φ then show that xy + x – y + 1 = 0
[Hint: Find xy + 1 and then show that x – y = –(xy + 1)]

If θ lies in the first quadrant and cosθ = `8/17`, then find the value of cos(30° + θ) + cos(45° – θ) + cos(120° – θ).

Find the value of the expression `cos^4  pi/8 + cos^4  (3pi)/8 + cos^4  (5pi)/8 + cos^4  (7pi)/8`

[Hint: Simplify the expression to `2(cos^4  pi/8 + cos^4  (3pi)/8) = 2[(cos^2  pi/8 + cos^2  (3pi)/8)^2 - 2cos^2  pi/8 cos^2  (3pi)/8]`

Find the general solution of the equation 5cos²θ + 7sin²θ – 6 = 0

Find the general solution of the equation sinx – 3sin2x + sin3x = cosx – 3cos2x + cos3x

Find the general solution of the equation `(sqrt(3) - 1) costheta + (sqrt(3) + 1) sin theta` = 2

[Hint: Put `sqrt(3) - 1` = r sinα, `sqrt(3) + 1` = r cosα which gives tanα = `tan(pi/4 - pi/6)` α = `pi/12`]

If sinθ + cosecθ = 2, then sin²θ + cosec²θ is equal to ______.

If f(x) = cos²x + sec²x, then ______.

[Hint: A.M ≥ G.M.]

If tanθ = `1/2` and tanΦ = `1/3`, then the value of θ + Φ is ______.

Which of the following is not correct?

The value of tan1° tan2° tan3° ... tan89° is ______.

The value of `(1 - tan^2 15^circ)/(1 + tan^2 15^circ)` is ______.

The value of cos1° cos2° cos3° ... cos179° is ______.

If tan θ = 3 and θ lies in third quadrant, then the value of sin θ  ______.

The value of tan 75° - cot 75° is equal to ______.

Which of the following is correct?

[Hint: 1 radian = `180^circ/pi = 57^circ30^'` approx]

If tanα = `m/(m +  1)`, tanβ = `1/(2m + 1)`, then α + β is equal to ______.

The minimum value of 3cosx + 4sinx + 8 is ______.

The value of tan3A - tan2A - tanA is equal to ______.

The value of sin(45° + θ) - cos(45° - θ) is ______.

The value of `cot(pi/4 + theta)cot(pi/4 - theta)` is ______.

cos2θ cos2Φ + sin²(θ – Φ) – sin²(θ + Φ) is equal to ______.

The value of cos12° + cos84° + cos156° + cos132° is ______.

If tanA = `1/2`, tanB = `1/3`, then tan(2A + B) is equal to ______.

The value of `sin  pi/10  sin  (13pi)/10` is ______.

`["Hint: Use"  sin18^circ = (sqrt5 - 1)/4 "and"  cos36^circ = (sqrt5 + 1)/4]`

The value of sin50° – sin70° + sin10° is equal to ______.

If sinθ + cosθ = 1, then the value of sin2θ is equal to ______.

If α + β = `pi/4`, then the value of (1 + tan α)(1 + tan β) is ______.

If sinθ = `(-4)/5` and θ lies in the third quadrant then the value of `cos  theta/2` is ______.

Number of solutions of the equation tan x + sec x = 2 cosx lying in the interval [0, 2π] is ______.

The value of `sin  pi/18 + sin  pi/9 + sin  (2pi)/9 + sin  (5pi)/18` is given by ______.

If A lies in the second quadrant and 3tanA + 4 = 0, then the value of 2cotA – 5cosA + sinA is equal to ______.

The value of cos²48° – sin²12° is ______.

[Hint: Use cos²A – sin² B = cos(A + B) cos(A – B)]

If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to ______.

If tanθ = `a/b`, then bcos2θ + asin2θ is equal to ______.

If for real values of x, cosθ = `x + 1/x`, then ______.

The value of `(sin 50^circ)/(sin 130^circ)` is ______.

If k = `sin(pi/18) sin((5pi)/18) sin((7pi)/18)`, then the numerical value of k is ______.

If tanA = `(1 - cos "B")/sin"B"`, then tan2A = ______.

If sinx + cosx = a, then sin⁶x + cos⁶x = ______.

If sinx + cosx = a, then |sinx – cosx| = ______.

In a triangle ABC with ∠C = 90° the equation whose roots are tan A and tan B is ______.

3(sinx – cosx)⁴ + 6(sinx + cosx)² + 4(sin⁶x + cos⁶x) = ______.

Given x > 0, the values of f(x) = `-3cos sqrt(3 + x + x^2)` lie in the interval ______.

The maximum distance of a point on the graph of the function y = `sqrt(3)` sinx + cosx from x-axis is ______.

State whether the statement is True or False? Also give justification.

If tanA = `(1 - cos B)/sinB`, then tan2A = tanB

State whether the statement is True or False? Also give justification.

The equality sinA + sin2A + sin3A = 3 holds for some real value of A.

State whether the statement is True or False? Also give justification.

Sin10° is greater than cos10°

State whether the statement is True or False? Also give justification.

`cos  (2pi)/15 cos  (4pi)/15 cos  (8pi)/15 cos  (16pi)/15 = 1/16`

State whether the statement is True or False? Also give justification.

One value of θ which satisfies the equation sin⁴θ - 2sin²θ - 1 lies between 0 and 2π.

State whether the statement is True or False? Also give justification.

If cosecx = 1 + cotx then x = 2nπ, 2nπ + `pi/2`

State whether the statement is True or False? Also give justification.

If tanθ + tan2θ + `sqrt(3)` tanθ tan2θ = `sqrt(3)`, then θ = `("n"pi)/3 + pi/9`

State whether the statement is True or False? Also give justification.

If tan(π cosθ) = cot(π sinθ), then `cos(theta - pi/4) = +- 1/(2sqrt(2))`.

In the following match each item given under the column C₁ to its correct answer given under the column C₂: