Samacheer Kalvi solutions for Mathematics - Volume 1 and 2 [English] Class 11 TN Board chapter 3 - Trigonometry [Latest edition]

Identify the quadrant in which an angle given measure lies
25°

Identify the quadrant in which an angle given measure lies
825°

Identify the quadrant in which an angle given measure lies
– 55°

Identify the quadrant in which an angle given measure lies
328°

Identify the quadrant in which an angle given measure lies
– 230°

For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
395°

For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
525°

For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
1150°

For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
– 270°

For each given angle, find a coterminal angle with measure of θ such that 0° ≤ θ < 360°
– 450°

If a cos θ − b sin θ = c, show that a sin θ + b cos θ = `+-  sqrt("a"^2 + "b"^2 - "c"^2)`

If sin θ + cos θ = m, show that cos⁶θ + sin⁶θ = `(4 - 3("m"^2 - 1)^2)/4`, where m² ≤ 2

If `(cos^4α)/(cos^2β) + (sin^4α)/(sin^2β)` = 1, prove that sin⁴α + sin⁴β = 2 sin²α sin²β

If `(cos^4alpha)/(cos^2beta) + (sin^4alpha)/(sin^2beta)` = 1, prove that `(cos^4beta)/(cos^2alpha) + (sin^4beta)/(sin^2alpha)` = 1

If y = `(2sinalpha)/(1 + cosalpha + sinalpha)`, then prove that `(1 - cosalpha + sinalpha)/(1 + sinalpha)` = y

If x = `sum_("n" = 0)^oo cos^(2"n") theta, y = sum_("n" = 0)^oo sin^(2"n") theta` and z = `sum_("n" = 0)^oo cos^(2"n") theta, sin^(2"n") theta, 0 < theta < pi/2`, then show that xyz = x + y + z. [Hint: Use the formula 1 + x + x² + x³ + . . . = `1/(1 - x), where |x| < 1]

If tan² θ = 1 – k², show that sec θ + tan³ θ cosec θ = (2 – k²)^3/2. Also, find the values of k for which this result holds

If sec θ + tan θ = p, obtain the values of sec θ, tan θ and sin θ in terms of p

If cot θ(1 + sin θ) = 4m and cot θ(1 – sin θ) = 4n then prove that (m² – n²)² = m

If cosec θ – sin θ = a³ and sec θ – cos θ = b³, then prove that a²b² (a² + b²) = 1

Eliminate θ from the equations a sec θ – c tan θ = b and b sec θ + d tan θ = c

Express the following angles in radian measure:
30°

Express the following angles in radian measure:
135°

Express the following angles in radian measure:
– 205°

Express the following angles in radian measure:
150°

Express the following angles in radian measure:
330°

Find the degree measure corresponding to the following radian measures
`pi/3`

Find the degree measure corresponding to the following radian measures
`pi/9`

Find the degree measure corresponding to the following radian measures
`(2pi)/5`

Find the degree measure corresponding to the following radian measures
`(7pi)/3`

Find the degree measure corresponding to the following radian measures
`(10pi)/9`

What must be the radius of a circular running path, around which an athlete must run 5 times in order to describe 1 km?

In a circle of diameter 40 cm, a chord is of length 20 cm. Find the length of the minor arc of the chord

Find the degree measure of the angle subtended at the centre of circle of radius 100 cm by an arc of length 22 cm

What is the length of the arc intercepted by a central angle of measure 41° in a circle of radius 10 ft?

If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii

The perimeter of a certain sector of a circle is equal to the length of the arc of a semi-circle having the same radius. Express the angle of the sector in degrees, minutes and seconds

An airplane propeller rotates 1000 times per minute. Find the number of degrees that a point on the edge of the propeller will rotate in 1 second

A train is moving on a circular track of 1500 m radius at the rate of 66 km/hr. What angle will it turn in 20 seconds?

A circular metallic plate of radius 8 cm and thickness 6 mm is melted and molded into a pie (a sector of the circle with thickness) of radius 16 cm and thickness 4 mm. Find the angle of the sector.

Find the values of sin(480°)

Find the values of sin (– 1110°)

Find the values of cos(300°)

Find the values of tan(1050°)

Find the values of cot(660°)

Find the values of `tan ((19pi)/3)`

Find the values of `sin (-(11pi)/3)`

`(5/7, (2sqrt(6))/7)` is a point on the terminal side of an angle θ in standard position. Determine the six trigonometric function values of angle θ

Find the value of the trigonometric functions for the following:
cos θ = `- 1/2`, θ lies in the III quadrant

Find the value of the trigonometric functions for the following:
cos θ = `2/3`, θ lies in the I quadrant

Find the value of the trigonometric functions for the following:
cos θ = `- 2/3`, θ lies in the IV quadrant

Find the value of the trigonometric functions for the following:
tan θ = −2, θ lies in the II quadrant

Find the value of the trigonometric functions for the following:
sec θ = `13/5`, θ lies in the IV quadrant

Prove that `(cot(180^circ + theta) sin(90^circ - theta) cos(- theta))/(sin(270^circ + theta) tan(- theta) "cosec"(360^circ + theta))` = cos²θ cotθ

Find all the angles between 0° and 360° which satisfy the equation sin²θ = `3/4`

Show that `sin^2  pi/18 + sin^2  pi/9 + sin^2  (7pi)/18 + sin^2  (4pi)/9` = 2

If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2` find the value of sin(x + y)

If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2`, find the value of cos(x − y)

If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2`, find the value of tan(x + y)

If sin A = `3/5` and cos B = `9/41 0 < "A" < pi/2, 0 < "B" < pi/2`, find the value of sin(A + B)

If sin A = `3/5` and cos B = `9/41, 0 < "A" < pi/2, 0 < "B" < pi/2`, find the value of cos(A – B)

Find cos(x − y), given that cos x = `- 4/5` with `pi < x < (3pi)/2`  and sin y = `- 24/25` with `pi < y < (3pi)/2`

Find sin(x – y), given that sin x = `8/17` with 0 < x < `pi/2`, and cos y = `- 24/25`, x < y < `(3pi)/2`

Find the value of cos 105°.

Find the value of sin105°.

Find the value of tan `(7pi)/12`

Prove that cos(30° + x) = `(sqrt(3) cos x - sin x)/2`

Prove that cos(π + θ) = − cos θ

Prove that sin(π + θ) = − sin θ.

Find a quadratic equation whose roots are sin 15° and cos 15°

Expand cos(A + B + C). Hence prove that cos A cos B cos C = sin A sin B cos C + sin B sin C cos A + sin C sin A cos B, if A + B + C = `pi/2`

Prove that sin(45° + θ) – sin(45° – θ) = `sqrt(2) sin θ`

Prove that sin(30° + θ) + cos(60° + θ) = cos θ

If a cos(x + y) = b cos(x − y), show that (a + b) tan x = (a − b) cot y

Prove that sin 105° + cos 105° = cos 45°

Prove that sin 75° – sin 15° = cos 105° + cos 15°

Show that tan 75° + cot 75° = 4

Prove that cos(A + B) cos C – cos(B + C) cos A = sin B sin(C – A)

Prove that sin(n + 1) θ sin(n – 1) θ + cos(n + 1) θ cos(n – 1)θ = cos 2θ, n ∈ Z

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

Prove that sin(A + B) sin(A – B) = sin²A – sin²B

Prove that cos(A + B) cos(A – B) = cos²A – sin²B = cos²B – sin²A

Prove that sin²(A + B) – sin²(A – B) = sin2A sin2B

Prove that cos 8θ cos 2θ = cos²5θ – sin²3θ

Show that cos² A + cos² B – 2 cos A cos B cos(A + B) = sin²(A + B)

If cos(α – β) + cos(β – γ) + cos(γ – α) = `- 3/2`, then prove that cos α + cos β + cos γ = sin α + sin β + sin γ = 0

Show that tan(45° + A) =  `(1 + tan"A")/(1 - tan"A")`

Show that tan(45° − A) = `(1 - tan "A")/(1 + tan "A")`

Prove that cot(A + B) = `(cot "A" cot "B" - 1)/(cot "A" + cot "B")`

If tan x = `"n"/("n" + 1)` and tan y = `1/(2"n" + 1)`, find tan(x + y)

Prove that `tan(pi/4 + theta) tan((3pi)/4 + theta)` = – 1

Find the value of tan(α + β), given that cot α = `1/2`, α ∈ `(pi, (3pi)/2)` and sec β = `- 5/3` β ∈ `(pi/2, pi)`

If θ + Φ = α and tan θ = k tan Φ, then prove that sin(θ – Φ) = `("k" - 1)/("k" + 1)` sin α

Find the value of cos 2A, A lies in the first quadrant, when cos A = `15/17`

Find the value of cos 2A, A lies in the first quadrant, when sin A = `4/5`

Find the value of cos 2A, A lies in the first quadrant, when tan A  `16/63`

If θ is an acute angle, then find `sin (pi/4 - theta/2)`, when sin θ = `1/25`

If θ is an acute angle, then find `cos (pi/4 + theta/2)`, when sin θ = `8/9`

If cos θ = `1/2 ("a" + 1/"a")`, show that cos 3θ = `1/2 ("a"^3 + 1/"a"^3)`

Prove that cos 5θ = 16 cos⁵θ – 20 cos³θ + 5 cos θ

Prove that sin 4α = `4 tan alpha (1 - tan^2alpha)/(1 + tan^2 alpha)^2`

If A + B = 45°, show that (1 + tan A)(1 + tan B) = 2

Prove that (1 + tan 1°)(1 + tan 2°)(1 + tan 3°) ..... (1 + tan 44°) is a multiple of 4

Prove that `tan (pi/4 + theta) - tan(pi/4 - theta)` = 2 tan 2θ

Show that `cot(7 1^circ/2) = sqrt(2) + sqrt(3) + sqrt(4) + sqrt(6)`

Prove that (1 + sec 2θ)(1 + sec 4θ) ... (1 + sec 2ⁿθ) = tan 2ⁿθ

Prove that `32(sqrt(3)) sin  pi/48  cos  pi/48  cos  pi/24  cos  pi/12  cos  pi/6` = 3

Express the following as a sum or difference
sin 35° cos 28°

Express the following as a sum or difference
sin 4x cos 2x

Express the following as a sum or difference
2 sin 10θ  cos 2θ

Express the following as a sum or difference
cos 5θ  cos 2θ

Express the following as a sum or difference
sin 5θ  sin 4θ

Express the following as a product
sin 75° sin 35°

Express the following as a product
cos 65° + cos 15°

Express the following as a product
sin 50° + sin 40°

Express the following as a product
cos 35° – cos 75°

Show that sin 12° sin 48° sin 54° = `1/8`

Show that `cos  pi/15  cos  (2pi)/15  cos  (3pi)/15  cos  (4pi)/15  cos  (5pi)/15  cos  (6pi)/15  cos  (7pi)/15 = 1/128`

Show that `(sin 8x cos x - sin 6x cos 3x)/(cos 2x cos x - sin 3x sin 4x)` = tan 2x

Show that `((cos theta -cos 3theta)(sin 8theta + sin 2theta))/((sin 5theta - sin theta) (cos 4theta - cos 6theta))` = 1

Prove that sin x + sin 2x + sin 3x = sin 2x (1 + 2 cos x)

Prove that `(sin 4x + sin 2x)/(cos 4x + cos 2x)` = tan 3x

Prove that 1 + cos 2x + cos 4x + cos 6x = 4 cos x cos 2x cos 3x

Prove that `sin  theta/2 sin  (7theta)/2 + sin  (3theta)/2 sin  (11theta)/2` =  sin 2θ sin 5θ

Prove that cos(30° – A) cos(30° + A) + cos(45° – A) cos(45° + A) = `cos 2"A" + 1/4`

Prove that `(sin x + sin 3x + sin 5x + sin 7x)/(cos x + cos x + cos 5x  cos 7x)` = tan 4x

Prove that `(sin(4"A" - 2"B") + sin(4"B" - 2"A"))/(cos(4"A" - 2"B") + cos(4"B" - 2"A"))` = tan(A + B)

Show that cot(A + 15°) – tan(A – 15°) = `(4cos2"A")/(1 + 2 sin2"A")`

If A + B + C = 180◦, prove that sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C

If A + B + C = 180°, prove that cos A + cos B − cos C = `- 1 + 4cos  "A"/2 cos  "B"/2 sin  "C"/2`

If A + B + C = 180°, prove that sin²A + sin²B + sin²C = 2 + 2 cos A cos B cos C

If A + B + C = 180°, prove that sin²A + sin²B − sin²C = 2 sin A sin B cos C

If A + B + C = 180°, prove that `tan  "A"/2  tan  "B"/2 + tan  "B"/2 tan  "C"/2 + tan  "C"/2 tan  "A"/2` = 1

If A + B + C = 180°, prove that sin A + sin B + sin C = `4 cos  "A"/2 cos  "B"/2 cos  "C"/2`

If A + B + C = 180°, prove that sin(B + C − A) + sin(C + A − B) + sin(A + B − C) = 4 sin A sin B sin C

If A + B + C = 2s, then prove that sin(s – A) sin(s – B)+ sin s  sin(s – C) = sin A sin B

If x + y + z = xyz, then prove that `(2x)/(1 - x^2) + (2y)/(1 - y^2) + (2z)/(1 - z^2) = (2x)/(1 - x^2) (2y)/(1 - y^2) (2z)/(1 - z^2)`

If A + B + C = `pi/2`, prove the following sin 2A + sin 2B + sin 2C = 4 cos A cos B cos C

If A + B + C = `pi/2`, prove the following cos 2A + cos 2B + cos 2C = 1 + 4 sin A sin B sin C

If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that cos² B + cos² C = 1

If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that sin² B + sin² C = 1

If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that cos B – cos C = `- 1 + 2sqrt(2) cos  "B"/2  sin  "C"/2`

Find the principal solution and general solution of the following:
sin θ = `-1/sqrt(2)`

Find the principal solution and general solution of the following:
cot θ = `sqrt(3)`

Find the principal solution and general solution of the following:
tan θ = `- 1/sqrt(3)`

Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°

sin⁴x = sin²x

Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°

2 cos²x + 1 = – 3 cos x

Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°

2 sin²x + 1 = 3 sin x

Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°

cos 2x = 1 − 3 sin x

Solve the following equations:
sin 5x − sin x = cos 3

Solve the following equations:
2 cos²θ + 3 sin θ – 3 = θ

Solve the following equations:
cos θ + cos 3θ = 2 cos 2θ

Solve the following equations:
sin θ + sin 3θ + sin 5θ = 0

Solve the following equations:
sin 2θ – cos 2θ – sin θ + cos θ = θ

Solve the following equations:
sin θ + cos θ = `sqrt(2)`

Solve the following equations:
`sin theta + sqrt(3) cos theta` = 1

Solve the following equations:
cot θ + cosec θ = `sqrt(3)`

Solve the following equations:
`tan theta + tan (theta + pi/3) + tan (theta + (2pi)/3) = sqrt(3)`

Solve the following equations:
cos 2θ = `(sqrt(5) + 1)/4`

Solve the following equations:
2cos ²x – 7 cos x + 3 = 0

In a ∆ABC, if `sin"A"/sin"C" = (sin("A" - "B"))/(sin("B" - "C"))` prove that a², b², C² are in Arithmetic Progression

The angles of a triangle ABC, are in Arithmetic Progression and if b : c = `sqrt(3) : sqrt(2)`, find ∠A

In a ∆ABC, if cos C = `sin "A"/(2sin"B")` show that the triangle is isosceles

In a ∆ABC, prove that `sin "B"/sin "C" = ("c" - "a"cos "B")/("b" - "a" cos"C")`

In an ∆ABC, prove that a cos A + b cos B + c cos C = 2a sin B sin C

In a ∆ABC, ∠A = 60°. Prove that b + c = `2"a" cos (("B" - "C")/2)`

In an ∆ABC, prove the following, `"a"sin ("A"/2 + "B") = ("b" + "c") sin  "A"/2`

In a ∆ABC, prove the following, a(cos B + cos C) = `2("b" + "c") sin^2  "A"/2`

In a ∆ABC, prove the following, `("a"^2 - "c"^2)/"b"^2 = (sin ("A" - "C"))/(sin("A" + "C"))`

In a ∆ABC, prove the following, `("a"sin("B" - "C"))/("b"^2 - "c"^2) = ("b"sin("C" - "A"))/("c"^2 - "a"^2) = ("c"sin("A" - "B"))/("a"^2 - "b"^2)`

In a ∆ABC, prove the following, `("a"+ "b")/("a" - "b") = tan(("A" + "B")/2) cot(("A" - "B")/2)`

In a ∆ABC, prove that (a² – b² + c²) tan B = (a² + b² – c²) tan C

An Engineer has to develop a triangular shaped park with a perimeter 120 m in a village. The park to be developed must be of maximum area. Find out the dimensions of the park

A rope of length 42 m is given. Find the largest area of the triangle formed by this rope and find the dimensions of the triangle so formed

Derive Projection formula from Law of sines

Derive Projection formula from Law of cosines

Determine whether the following measurements produce one triangle, two triangles or no triangle:
∠B = 88°, a = 23, b = 2. Solve if solution exists

If the sides of a ∆ABC are a = 4, b = 6 and c = 8, then show that 4 cos B + 3 cos C = 2

In a ∆ABC, if a = `sqrt(3) - 1`, b = `sqrt(3) + 1` and C = 60° find the other side and other two angles

In any ∆ABC, prove that the area ∆ = `("b"^2 + "c"^2 - "a"^2)/(4 cot "A")`

In a ∆ABC, if a = 12 cm, b = 8 cm and C = 30°, then show that its area is 24 sq.cm

In a ∆ABC, if a = 18 cm, b = 24 cm and c = 30 cm, then show that its area is 216 sq.cm

Two soldiers A and B in two different underground bunkers on a straight road, spot an intruder at the top of a hill. The angle of elevation of the intruder from A and B to the ground level in the eastern direction are 30° and 45° respectively. If A and B stand 5km apart, find the distance of the intruder from B

A researcher wants to determine the width of a pond from east to west, which cannot be done by actual measurement. From a point P, he finds the distance to the eastern-most point of the pond to be 8 km, while the distance to the westernmost point from P to be 6 km. If the angle between the two lines of sight is 60°, find the width of the pond

Two Navy helicopters A and B are flying over the Bay of Bengal at saine altitude from sea level to search a missing boat. Pilots of both the helicopters sight the boat at the same time while they are apart 10 km from each other. If the distance of the boat from A is 6 km and if the line segment AB subtends 60° at the boat, find the distance of the boat from B

A straight tunnel is to be made through a mountain. A surveyor observes the two extremities A and B of the tunnel to be built from a point P in front of the mountain. If AP = 3 km, BP = 5 km, and ∠APB = 120°, then find the length of the tunnel to be built

A farmer wants to purchase a triangular-shaped land with sides 120 feet and 60 feet and the angle included between these two sides is 60°. If the land costs Rs.500 per square feet, find the amount he needed to purchase the land. Also, find the perimeter of the land

A fighter jet has to hit a small target by flying a horizontal distance. When the target is sighted, the pilot measures the angle of depression to be 30°. If after 100 km, the target has an angle of depression of 45°, how far is the target from the fighter jet at that instant?

A plane is 1 km from one landmark and 2 km from another. From the plane’s point of view, the land between them subtends an angle of 45°. How far apart are the landmarks?

A man starts his morning walk at a point A reaches two points B and C and finally back to A such that ∠A = 60° and ∠B = 45°, AC = 4 km in the ∆ABC. Find the total distance he covered during his morning walk

Two vehicles leave the same place P at the same time moving along two different roads. One vehicle moves at an average speed of 60 km/hr and the other vehicle moves at an average speed of 80 km/hr. After half an hour the vehicle reaches destinations A and B. If AB subtends 60° at the initial point P, then find AB

Suppose that a satellite in space, an earth station, and the centre of earth all lie in the same plane. Let r be the radius of earth and R he the distance from the centre of earth to the satellite. Let d be the distance from the earth station to the satellite. Let 30° be the angle of elevation from the earth station to the satellite, If the line segment connecting the earth station and satellite subtends angle α at the centre of earth then prove that d = `"R"sqrt(1 + ("r"/"R")^2 - 2 ("r"/"R") cos alpha)`

Find the principal value of `sin^-1  1/sqrt(2)`

Find the principal value of `cos^-1  sqrt(3)/2`

Find the principal value of cosec^–1(– 1)

Find the principal value of `sec^-1 (- sqrt(2))`.

Find the principal value of `tan^-1 (sqrt(3))`

A man standing directly opposite to one side of a road of width x meter views a circular shaped traffic green signal of diameter ‘a’ meter on the other side of the road. The bottom of the green signal Is ‘b’ meter height from the horizontal level of viewer’s eye. If ‘a’ denotes the angle subtended by the diameter of the green signal at the viewer’s eye, then prove that α = `tan^-1 (("a" + "b")/x) - tan^-1 ("b"/x)`

Choose the correct alternative:
`1/(cos 80^circ) - sqrt(3)/(sin 80^circ)` = 

Choose the correct alternative:
If cos 28° + sin 28° = k³, then cos 17° is equal to

Choose the correct alternative:
The maximum value of `4sin^2x + 3cos^2x + sin   x/2 + cos  x/2` is

Choose the correct alternative:
`(1 + cos  pi/8) (1 + cos  (3pi)/8) (1 + cos  (5pi)/8) (1 + cos  (7pi)/8)` =

Choose the correct alternative:
If `pi < 2theta < (3pi)/2`, then `sqrt(2 + sqrt(2 + 2cos4theta)` equals to

Choose the correct alternative:
If tan 40° = λ, then `(tan 140^circ - tan 130^circ)/(1 + tan 140^circ *  tan 130^circ)` =

Choose the correct alternative:
cos 1° + cos 2° + cos 3° + ... + cos 179° =

Choose the correct alternative:
Let fk(x) = `1/"k" [sin^"k" x + cos^"k" x]` where x ∈ R and k ≥ 1. Then f₄(x) − f₆(x) = 

Choose the correct alternative:
Which of the following is not true?

Choose the correct alternative:
cos 2θ cos 2ϕ+ sin² (θ – ϕ) – sin² (θ + ϕ) is equal to

Choose the correct alternative:
`(sin("A" - "B"))/(cos"A" cos"B") + (sin("B" - "C"))/(cos"B" cos"C") + (sin("C" - "A"))/(cos"C" cos"A")` is 

Choose the correct alternative:
If cos pθ + cos qθ = 0 and if p ≠ q, then θ is equal to (n is any integer)

Choose the correct alternative:
If tan α and tan β are the roots of x² + ax + b = 0 then `(sin(alpha + beta))/(sin alpha sin beta)` is equal to

Choose the correct alternative:
In a triangle ABC, sin²A + sin²B + sin²C = 2, then the triangle is

Choose the correct alternative:
If f(θ) = |sin θ| + |cos θ| , θ ∈ R, then f(θ) is in the interval

Choose the correct alternative:
`(cos 6x + 6 cos 4x + 15cos x + 10)/(cos 5x + 5cs 3x + 10 cos x)` is equal to

Choose the correct alternative:
The triangle of maximum area with constant perimeter 12m

Choose the correct alternative:
A wheel is spinning at 2 radians/second. How many seconds will it take to make 10 complete rotations?

Choose the correct alternative:
If sin α + cos α = b, then sin 2α is equal to

Choose the correct alternative:
In a ∆ABC, if
(i) `sin  "A"/2 sin  "B"/2 sin  "C"/2 > 0`
(ii) sin A sin B sin C > 0 then