Frank solutions for Mathematics [English] Class 9 ICSE chapter 27 - Trigonometrical Ratios of Standard Angles [Latest edition]

Without using tables, evaluate the following: sin60° sin30°+ cos30° cos60°

Without using tables, evaluate the following: sec30° cosec60° + cos60° sin30°.

Without using tables, evaluate the following sec45° sin45° - sin30° sec60°.

Without using tables, evaluate the following: sin²30° sin²45° + sin²60° sin²90°.

Without using tables, evaluate the following: tan²30° + tan²60° + tan²45°

Without using tables, evaluate the following: sin²30° cos²45° + 4tan²30° + sin²90° + cos²0°

Without using tables, evaluate the following: cosec²45° sec²30° - sin²30° - 4cot²45° + sec²60°.

Without using tables, evaluate the following: cosec³30° cos60° tan³45° sin²90° sec²45° cot30°.

Without using tables, evaluate the following: (sin90° + sin45° + sin30°)(sin90° - cos45° + cos60°).

Without using tables, evaluate the following: 4(sin⁴30° + cos⁴60°) - 3(cos²45° - sin²90°).

Without using table, find the value of the following:

${\displaystyle \frac{\sin 30°-\sin 90°+ 2\cos 0°}{\tan 30°\tan 60°}}$ 

Without using tables, find the value of the following: ${\displaystyle \frac{\sin 30°}{\sin 45°}+\frac{\tan 45°}{\sec 60°}-\frac{\sin 60°}{\cot 45°}-\frac{\cos 30°}{\sin 90°}}$

Without using tables, find the value of the following: ${\displaystyle \frac{\tan 45°}{\text{cosec}30°}+\frac{\sec 60°}{\cot 45°}-\frac{5\sin 90°}{2\cos 0°}}$

Without using table, find the value of the following: ${\displaystyle \frac{{\tan }^{2}60°+4{\cos }^{2}45°+3{\sec }^{2}30°+5\cos 90°}{\cos ec30°+\sec 60°-{\cot }^{2}30°}}$

Without using tables, find the value of the following: ${\displaystyle \frac{4}{{\cot }^{2}30°}+\frac{1}{{\sin }^{2}60°}-{\cos }^{2}45°}$

Prove that: sin60°. cos30° - sin60^°. sin30^° = ${\displaystyle \frac{1}{2}}$

Prove that : cos60° . cos30° - sin60° . sin30° = 0

Prove that : sec²45° - tan²45° = 1

Prove that: ${\displaystyle {\left(\frac{\cot 30°+1}{\cot 30°-1}\right)}^2=\frac{\sec 30°+1}{\sec 30°-1}}$

Find the value of 'A', if 2 cos A = 1

Find the value of 'A', if 2 sin 2A = 1

Find the value of 'A', if cosec 3A = ${\displaystyle \frac{2}{\sqrt{3}}}$

Find the value of 'A', if 2cos 3A = 1

Find the value of 'A', if ${\displaystyle \sqrt{3}\cot \text{A}}$ = 1

Find the value of 'A', if cot 3A = 1

Find the value of 'A', if (1 - cosec A)(2 - sec A) = 0

Find the value of 'A', if (2 - cosec 2A) cos 3A = 0

If sin α + cosβ = 1 and α= 90°, find the value of 'β'.

Solve for 'θ': ${\displaystyle \sin \frac{\theta }{3}}$ = 1

Solve for 'θ': cot²(θ - 5)° = 3

Solve for 'θ': ${\displaystyle \sec (\frac{\theta }{2}+10°)=\frac{2}{\sqrt{3}}}$

Find the value of x in the following:  2 sin3x = ${\displaystyle \sqrt{3}}$

Find the value of x in the following: ${\displaystyle 2\sin \frac{x}{2}}$ = 1

Find the value of x in the following: ${\displaystyle \sqrt{3}\sin x}$ = cos x

Find the value of x in the following: tan x = sin45° cos45° + sin30°

Find the value of x in the following: ${\displaystyle \sqrt{3}}$tan 2x = cos60° + sin45° cos45°

Find the value of x in the following: cos2x = cos60° cos30° + sin60° sin30°

If sinθ = cosθ and 0° < θ<90°, find the value of 'θ'.

If tanθ= cotθ and 0°≤ θ ≤ 90°, find the value of 'θ'.

If ${\displaystyle \sqrt{2}=1.414\text{and}\sqrt{3}=1.732}$, find the value of the following correct to two decimal places tan60°

If θ = 30°, verify that: tan2θ = ${\displaystyle \frac{2\tan \theta }{1-{\tan }^2\theta }}$

If θ = 30°, verify that: sin2θ = ${\displaystyle \frac{2\tan \theta }{1++{\tan }^2\theta }}$

If A = 30°, verify that cos2θ = ${\displaystyle \frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }}$ = cos⁴θ - sin⁴θ = 2cos²θ - 1 - 2sin²θ

If θ = 30°, verify that: sin 3θ = 4sinθ . sin(60° - θ) sin(60° + θ)

If θ = 30°, verify that: 1 - sin 2θ = (sinθ - cosθ)² 

Evaluate the following: ${\displaystyle \frac{(\sin 3\theta -2\sin 4\theta )}{(\cos 3\theta -2\cos 4\theta )}}$ when 2θ = 30°

Evaluate the following: ${\displaystyle \frac{(1-\cos \theta )(1+\cos \theta )}{(1-\sin \theta )(1+\sin \theta )}}$ if θ = 30°

If θ = 15°, find the value of: cos3θ - sin6θ + 3sin(5θ + 15°) - 2 tan²3θ

If A = B = 60°, verify that: cos(A - B) = cosA cosB + sinA sinB

If A = B = 60°, verify that: sin(A - B) = sinA cosB - cosA sinB

If A = B = 60°, verify that: tan(A - B) = ${\displaystyle \frac{\tan \text{A}-\tan \text{B}}{1+\tan \text{A}\tan \text{B}}}$

If A = 30° and B = 60°, verify that: sin (A + B) = sin A cos B + cos A sin B

If A = 30° and B = 60°, verify that: cos (A + B) = cos A cos B - sin A sin B

If A = 30° and B = 60°, verify that: ${\displaystyle \frac{\sin (\text{A}+\text{B})}{\cos \text{A}.\cos \text{B}}}$ = tanA + tanB

If A = 30° and B = 60°, verify that: ${\displaystyle \frac{\sin (\text{A}-\text{B})}{\sin \text{A}.\sin \text{B}}}$ = cotB - cotA

If A = B = 45°, verify that sin (A - B) = sin A .cos B - cos A.sin B

If A = B = 45°, verify that cos (A − B) = cos A. cos B + sin A. sin B

If sin(A - B) = sinA cosB - cosA sinB and cos(A - B) = cosA cosB + sinA sinB, find the values of sin15° and cos15°.

If θ < 90°, find the value of: sin²θ + cos²θ

If θ < 90°, find the value of: ${\displaystyle {\tan }^2\theta -\frac{1}{{\cos }^2\theta }}$

If ${\displaystyle \sqrt{3}}$sec 2θ = 2 and θ< 90°, find the value of θ

If ${\displaystyle \sqrt{3}}$ sec 2θ = 2 and θ< 90°, find the value of

cos 3θ

If ${\displaystyle \sqrt{3}}$ sec 2θ = 2 and θ< 90°, find the value of

cos² (30° + θ) + sin² (45° - θ)

In the given figure, PQ = 6 cm, RQ = x cm and RP = 10 cm, find
a. cosθ
b. sin²θ- cos²θ
c. Use tanθ to find the value of RQ

Find the value of: ${\displaystyle \sqrt{\frac{1-{\sin }^{2}60°}{1+{\sin }^{2}60°}}}$ If 3 tan²θ - 1 = 0, find the value 
a. cosθ
b. sinθ

If sin(A +B) = 1(A -B) = 1, find A and B.

If tan(A - B) = ${\displaystyle \frac{1}{\sqrt{3}}}$ and tan(A + B) = ${\displaystyle \sqrt{3}}$, find A and B.

If sin(A - B) = ${\displaystyle \frac{1}{2}}$ and cos(A + B) = ${\displaystyle \frac{1}{2}}$, find A and B.

In ΔABC right angled at B, ∠A = ∠C. Find the value of:
(i) sinA cosC + cosA sinC
(ii) sinA sinB + cosA cosB

If tan ${\displaystyle \text{A}=\frac{1}{2},\tan \text{B}=\frac{1}{3}\text{and}\tan (\text{A}+\text{B})=\frac{\tan \text{A}+\tan \text{B}}{1-\tan \text{A}\tan \text{B}}}$, find A + B.

Find the value of 'x' in each of the following:

Find the value of 'x' in each of the following:

Find the value of 'x' in each of the following:

Find the value of 'x' in each of the following:

Find the length of AD. Given: ∠ABC = 60°, ∠DBC = 45° and BC = 24 cm.

Find lengths of diagonals AC and BD. Given AB = 24 cm and ∠BAD = 60°.

In a trapezium ABCD, as shown, AB ‖ DC, AD = DC = BC = 24 cm and ∠A = 30°. Find: length of AB

Find the length of EC.

In the given figure, AB and EC are parallel to each other. Sides AD and BC are 1.5 cm each and are perpendicular to AB. Given that ∠AED = 45° and ∠ACD = 30°. Find:
a. AB
b. AC
c. AE

In the given figure, ∠B = 60°, ∠C = 30°, AB = 8 cm and BC = 24 cm. Find:
a. BE
b. AC

Find:
a. BC
b. AD
c. AC

Find the value 'x', if:

Find the value 'x', if:

Find the value 'x', if:

Find the value 'x', if:

Find the value 'x', if:

Find the value 'x', if:

Find the value of 'y' if ${\displaystyle \sqrt{3}}$ = 1.723.
Given your answer correct to 2 decimal places.

Find the value of 'y' if ${\displaystyle \sqrt{3}}$ = 1.723.
Given your answer correct to 2 decimal places.

In the given figure, if tan θ = ${\displaystyle \frac{5}{13},\tan \alpha =\frac{3}{5}}$ and RS = 12m, find the value of 'h'.

Find x and y, in each of the following figure:

Find x and y, in each of the following figure:

If tan x° = ${\displaystyle \frac{5}{12}.\tan y°=\frac{3}{4}}$ and AB = 48m; find the length CD.

In a right triangle ABC, right angled at C, if ∠B = 60° and AB = 15units, find the remaining angles and sides.

If ΔABC is a right triangle such that ∠C = 90°, ∠A = 45° and BC =7units, find ∠B, AB and AC.

In a rectangle ABCD, AB = 20cm, ∠BAC = 60°, calculate side BC and diagonals AC and BD.

In right-angled triangle ABC; ∠B = 90°. Find the magnitude of angle A, if:
a. AB is ${\displaystyle \sqrt{3}}$ times of BC.
B. BC is ${\displaystyle \sqrt{3}}$ times of BC.

A ladder is placed against a vertical tower. If the ladder makes an angle of 30° with the ground and reaches upto a height of 18 m of the tower; find length of the ladder.

The perimeter of a rhombus is 100 cm and obtuse angle of it is 120°. Find the lengths of its diagonals.

In the given figure; ∠B = 90°, ∠ADB = 30°, ∠ACB = 45° and AB = 24 m. Find the length of CD.

In the given figure, a rocket is fired vertically upwards from its launching pad P. It first rises 20 km vertically upwards and then 20 km at 60° to the vertical. PQ represents the first stage of the journey and QR the second. S is a point vertically below R on the horizontal level as P, find:
a. the height of the rocket when it is at point R.
b. the horizontal distance of point S from P.

Evaluate the following: ${\displaystyle \frac{\sin 62°}{\cos 28°}}$

Evaluate the following: ${\displaystyle \frac{\sec 34°}{\text{cosec}56°}}$

Evaluate the following: ${\displaystyle \frac{\tan 12°}{\cot 78°}}$

Evaluate the following: ${\displaystyle \frac{\sin 25°\cos 43°}{\sin 47°\cos 65°}}$

Evaluate the following: ${\displaystyle \frac{\sec 32°\cot 26°}{\tan 64°\text{cosec}58°}}$

Evaluate the following: ${\displaystyle \frac{\cos 34°\cos 35°}{\sin 57°\sin 56°}}$

Evaluate the following: sin31° - cos59°

Evaluate the following: cot27° - tan63°

Evaluate the following: cosec 54° - sec 36°

Evaluate the following: sin28° sec62° + tan49° tan41°

Evaluate the following: sec16° tan28° - cot62° cosec74°

Evaluate the following: sin22° cos44° - sin46° cos68°

Evaluate the following: ${\displaystyle \frac{\sin 36°}{\cos 54°}+\frac{\sec 31°}{\text{cosec}59°}}$

Evaluate the following: ${\displaystyle \frac{\tan 42°}{\cot 48°}+\frac{\cos 33°}{\sin 57°}}$

Evaluate the following: ${\displaystyle \frac{2\sin 28°}{\cos 62°}+\frac{3\cot 49°}{\tan 41°}}$

Evaluate the following: ${\displaystyle \frac{5\sec 68°}{\text{cosec}22°}+\frac{3\sin 52°\sec 38°}{\cot 51°\cot 39°}}$

Express each of the following in terms of trigonometric ratios of angles between 0° and 45°: sin65° + cot59°

Express each of the following in terms of trigonometric ratios of angles between 0° and 45°: cos72° - cos88°

Express each of the following in terms of trigonometric ratios of angles between 0° and 45°: cosec64° + sec70°

Express each of the following in terms of trigonometric ratios of angles between 0° and 45°: tan77° - cot63° + sin57°

Express each of the following in terms of trigonometric ratios of angles between 0° and 45°: sin53° + sec66° - sin50°

Express each of the following in terms of trigonometric ratios of angles between 0° and 45°: cos84° + cosec69° - cot68°

Evaluate the following: sin35° sin45° sec55° sec45°

Evaluate the following: cot20° cot40° cot45° cot50° cot70°

Evaluate the following: cos39° cos48° cos60° cosec42° cosec51°

Evaluate the following: sin(35° + θ) - cos(55° - θ) - tan(42° + θ) + cot(48° - θ)

Evaluate the following: tan(78° + θ) + cosec(42° + θ) - cot(12° - θ) - sec(48° - θ)

Evaluate the following: ${\displaystyle \frac{3\sin 37°}{\cos 53°}-\frac{5\text{cosec}39°}{\sec 51°}+\frac{4\tan 23°\tan 37°\tan 67°\tan 53°}{\cos 17°\cos 67°\text{cosec}73°\text{cosec}23°}}$

Evaluate the following: ${\displaystyle \frac{\sin 0°\sin 35°\sin 55°\sin 75°}{\cos 22°\cos 64°\cos 58°\cos 90°}}$

Evaluate the following: ${\displaystyle \frac{2\sin 25°\sin 35°\sec 55°\sec 65°}{5\tan 29°\tan 45°\tan 61°}+\frac{3\cos 20°\cos 50°\cot 70°\cot 40°}{5\tan 20°\tan 50°\sin 70°\sin 40°}}$

Evaluate the following: ${\displaystyle \frac{3{\sin }^{2}40°}{4{\cos }^{2}50°}-\frac{{\text{cosec}}^{2}28°}{4{\sec }^{2}62°}+\frac{\cos 10°\cos 25°\cos 45°\text{cosec}80°}{2\sin 15°\sin 25°\sin 45°\sin 65°\sec 75°}}$

Evaluate the following: ${\displaystyle \frac{5\cot 5°\cot 15°\cot 25°\cot 35°\cot 45°}{7\tan 45°\tan 55°\tan 65°\tan 75°\tan 85°}+\frac{2\text{cosec}12°\text{cosec}24°\cos 78°\cos 66°}{7\sin 14°\sin 23°\sec 76°\sec 67°}}$

If cos3θ = sin(θ - 34°), find the value of θ if 3θ is an acute angle.

If tan4θ = cot(θ + 20°), find the value of θ if 4θ is an acute angle.

If sec2θ = cosec3θ, find the value of θ if it is known that both 2θ and 3θ are acute angles.

If sin(θ - 15°) = cos(θ - 25°), find the value of θ if (θ-15°) and (θ - 25°) are acute angles.

If A, B and C are interior angles of ΔABC, prove that sin${\displaystyle \left(\frac{\text{A}+\text{B}}{2}\right)=\cos \frac{\text{C}}{2}}$

If P, Q and R are the interior angles of ΔPQR, prove that ${\displaystyle \cot \left(\frac{\text{Q}+\text{R}}{2}\right)=\tan \frac{\text{P}}{2}}$

If cosθ = sin60° and θ is an acute angle find the value of 1- 2 sin²θ

If secθ= cosec30° and θ is an acute angle, find the value of 4 sin²θ - 2 cos²θ.

Prove the following: tanθ tan(90° - θ) = cotθ cot(90° - θ)

Prove the following: sin58° sec32° + cos58° cosec32° = 2

Prove the following: ${\displaystyle \frac{\tan (90°-\theta )\cot \theta }{{\text{cosec}}^2\theta }}$ = cos²θ

Prove the following: sin²30° + cos²30° = ${\displaystyle \frac{1}{2}\sec 60°}$

If A + B = 90°, prove that ${\displaystyle \frac{\tan \text{A}\tan \text{B}+\tan \text{A}\cot \text{B}}{\sin \text{A}\sec \text{B}}-\frac{{\sin }^2\text{B}}{{\cos }^2\text{A}}}$ = tan²A