Trigonometrical Ratios Table

The Trigonometric Ratios Table shows the exact values of sin, cos, tan, cosec, sec, and cot for the standard angles 0°, 30°, 45°, 60°, and 90°. It is one of the most important reference tables in mathematics because it helps students solve trigonometry problems quickly without lengthy calculations. This table is widely used by students in middle school, high school, engineering, physics, and competitive examinations. Learning these values makes it easier to solve triangles, heights and distances, and many real-life applications of trigonometry.

Table of Contents:

What is Trigonometric Ratios Table?

Standard Angle Table

How to Read the Table?

Meaning of Each Ratio

Easy Memory Trick

Solved Examples

Practice Questions

Worksheet

Game

Quiz

PDF Download

Frequently Asked Questions

What is Trigonometrical Ratios Table?

A trigonometrical ratios table shows the values of sin, cos, tan, cosec, sec, and cot for standard angles.

The standard angles of trigonometrical ratios are 0°, 30°, 45°, 60° and 90°.

The values of trigonometrical ratios of standard angles are very important to solve the trigonometrical problems. Therefore, it is necessary to remember the value of the trigonometrical ratios of these standard angles. The sine, cosine and tangent of the standard angles are given below in the table.

Meaning of Each Trigonometric Ratio:

In a right triangle, the six trigonometric ratios compare the lengths of the sides. These ratios help us find unknown sides and angles.

Trigonometric Table in Sexagesimal System:

#Free Download sin, cos, tan Table 

Free Download sin cos tan Chart

Trigonometric Table in Circular System:

Trigonometry Table Free Download

Free Download Trigonometric Ratios Table

Note: Values of sin θ and cos θ lie between 0 and 1 (both inclusive)

How to Read the Trigonometric Ratios Table?

Reading the trigonometric ratios table is simple. Follow these steps:

Step I: Choose the Angle

Look at the top row to find the required angle.

For Example: If the angle is 30°, go to the 30° column.

Step II: Choose the Trigonometric Ratio

Look at the left column to find the required trigonometric ratio.

The six ratios are:

• sin (sine)
• cos (cosine)
• tan (tangent)
• csc (cosecant)
• sec (secant)
• cot (cotangent)

Step III: Find the Value

The value is found where the row and column meet.

Examples:

Note:

• Top Row → Angles (0°, 30°, 45°, 60°, 90°)
• Left Column → Trigonometric Ratios (sin, cos, tan, csc, sec, cot)
• Intersection → Exact value of the required ratio.

Solved Example:

1. How to find the exact value tan 60° from Trigonometric Ratios Table?

Solution:

Step I: Find 60° in the top row.

Step II: Find tan in the left column.

Step III: The value at the intersection is √3.

Therefore, the exact value of tan 60° is √3.

This same method works for every value in the trigonometric ratios table.

Interactive Calculator of Trigonometric Ratios Table of Standard Angles:

Trigonometric Ratios Calculator

Select a standard angle to view exact and decimal values for all six trigonometric ratios.

Choose a standard angle 0°30°45°60°90°

The calculator shows simplified exact forms alongside approximations to four decimal places.

Trigonometric values for the selected standard angle

Ratio	Exact value	Decimal value

Copy values Reset to 0°

Sin Cos Tan Wheel:

Download Free Trig Table PDF:

Trigonometric Ratios Table: Drag-and-Drop Game

Trigonometric Ratios Table: Drag-and-Drop Game

Match each ratio with its exact standard-angle value.

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Drag the correct answer into the answer area. On a phone or tablet, tap a value first, then tap the answer area.

Drop or place your answer here Choose a value, then activate this answer area to submit it.

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How to Play: Trigonometry Table Drag-and-Drop ChallengeMaster your trigonometric values with this fast-paced interactive matching game! Test your memory and see how quickly you can fill the table.

🎯 Game Objective

Drag the correct trigonometric values from the answer bank and drop them into their corresponding spots on the ratio table before the timer runs out.

📋 Step-by-Step Instructions:

Step I: Pick a Value — Look at the available number blocks in the answer bank at the bottom of the game area.

Step II: Match the Ratio — Find the empty cell in the grid where that value mathematically belongs (matching the Trigonometric Ratio with the degree angle).

Step III: Drag and Drop — Click and hold (or tap and hold on mobile) to drag your selected block into the correct target box.

Step IV: Check Your Accuracy — Correct drops will lock directly into place. Wrong answers will snap back to the pool automatically.

Step V: Beat the Clock — Complete the full matrix as fast as possible to unlock a perfect score and challenge your personal best time!

Easy Trick to Remember Trigonometric Values:

Memorization Tricks for Trigonometric Ratios Table:

Learning the trigonometric table doesn't require memorizing dozens of numbers. You can easily construct the entire sine and cosine table using a simple step-by-step counting trick.

To remember the above values:

Step I: Write down numbers from 0 to 4

For the standard angles (0°, 30°, 45°, 60° and 90°), write down consecutive integers starting from 0 up to 4:

          0     1     2     3     4

Step II: Divide each number by 4

          \(\frac{0}{4}\),  \(\frac{1}{4}\),  \(\frac{2}{4}\),  \(\frac{3}{4}\),  \(\frac{4}{4}\)

Step III: Take the positive square root of each fraction

sin 0° = \(\sqrt{\frac{0}{4}}\) = 0

sin 30° = \(\sqrt{\frac{1}{4}}\) = \(\frac{1}{2}\)

sin 45° = \(\sqrt{\frac{2}{4}}\) = \(\frac{1}{\sqrt{2}}\) = \(\frac{\sqrt{2}}{2}\)

sin 60° = \(\sqrt{\frac{3}{4}}\) = \(\frac{\sqrt{3}}{2}\)

sin 90° = \(\sqrt{\frac{4}{4}}\) = 1.

Step IV: Reverse the order for Cosine (cos θ)

To get the values for cos θ, simply write the sine values in reverse order (start from 90° and move down to 0°):

cos 0° = sin 90° = 1

cos 30° = sin 60° = \(\frac{\sqrt{3}}{2}\)

cos 45° = sin 45° = \(\frac{\sqrt{2}}{2}\)

cos 60° = sin 30° = \(\frac{1}{2}\)

cos 90° = sin 0° = 0.

Note: Write the values of sin 0°, sin 30°, sin 45°, sin 60° and sin 90° in reverse order and get the values of cos 0°, cos 30°, cos 45°, cos 60° and cos 90° respectively.

If θ be an acute angle, the values of sin θ and cos θ lie between 0 and 1 (both inclusive).

The sine values of the standard angles 0°, 30°, 45°, 60° and 90° are respectively the positive square roots of \(\frac{0}{4}\),  \(\frac{1}{4}\),  \(\frac{2}{4}\),  \(\frac{3}{4}\), and \(\frac{4}{4}\).

Since we know the sin and cos values of the standard angles from the table, we can easily find the values of the other trigonometric ratios of standard angles. 

Using basic reciprocal relationships (csc θ = \(\frac{1}{sin θ}\), sec θ = \(\frac{1}{cos θ}\), cot θ = \(\frac{1}{tan θ}\) and quotient rules tan θ = \(\frac{sin θ}{cos θ}\), we find the values for all six standard functions:

The cosecant or cosec of the standard angles 0°, 30°, 45°, 60° and 90°:

csc 0° = not defined.

csc 30° = 2

csc 45° = √2

csc 60° = \(\frac{2\sqrt{3}}{3}\)

csc 90° = 1.

The secant of the standard angles 0°, 30°, 45°, 60° and 90°:

sec 0° = 1

sec 30° = \(\frac{2\sqrt{3}}{3}\)

sec 45° = √2

sec 60° = 2

sec 90° = not defined.

The cotangent of the standard angles 0°, 30°, 45°, 60° and 90°:

cot 0° = not defined.

cot 30° = √3

cot 45° = 1

cot 60° = \(\frac{\sqrt{3}}{3}\)

cot 90° = 0

The tangent of the standard angles 0°, 30°, 45°, 60° and 90°:

tan 0° = 0

tan 30° = \(\frac{\sqrt{3}}{3}\)

tan 45° = 1

tan 60° = √3

tan 90° = not defined.

Quick Summary Table of Ratios:

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Remember: Values for csc, sec, and cot can be derived using the reciprocal identities above.

• sin θ values increase from 0 to 1.
• cos θ values decrease from 1 to 0.
• sin θ = cos θ at 45°.

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Note:

(i) For sin values: Start from 0 and go up.

(ii) For cos values: Start from 1 and go down.

     This is why sin increases and cos decreases.

(iii) sin values go up, cos values go down

(iv) At 45°, both are equal. This helps you avoid confusion.

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Quiz on Trigonometric Ratios Table

SOH CAH TOA Poster – Trigonometric Ratios Triangle Chart:

Learn SOH CAH TOA easily with this colorful trigonometric ratios poster showing the sine, cosine and tangent formulas for a right triangle.

Solved Examples on Trigonometrical Ratios Table:

1. If the csc of an angle complementary to A be \(\frac{2√3}{3}\), find tan A.

Solution:

csc (Complementary of ∠A) = \(\frac{2√3}{3}\)

[Using trigonometric ratio table, the value of csc 60° = \(\frac{2√3}{3}\)]

csc (Complementary of ∠A) = csc 60°

Complementary of ∠A = 60°

∠A = (90° - 60°)

∠A = 30°

Therefore, tan A = tan 30°

                        = \(\frac{√3}{3}\), [From Trig Ratios Table]

2. If A = 30° then show that, 2 sin A cos A = sin 2A

Solution:

L.H.S. = 2 sin A cos A

          = 2 sin 30° cos 30°

[From the Trigonometrical Ratios Table, we get  sin 30° = \(\frac{1}{2}\) and cos 30° = \(\frac{√3}{2}\)]

          = 2 × \(\frac{1}{2}\) × \(\frac{√3}{2}\)

          = \(\frac{√3}{2}\)

R.H.S. = sin 2A

          = sin (2 × 30°)

          = sin 60°

[From the Trigonometrical Ratios Table, we get  sin 60° = \(\frac{√3}{2}\)]

          = \(\frac{√3}{2}\)

Therefore, L.H.S. = R.H.S.

3. If A = 30° then show that, tan 2A = \(\frac{2 tan A}{1 - tan^{2} A}\).

Solution:

L.H.S. = tan 2A

          = tan (2 × 30°)

          = tan 60°

[From the Trigonometrical Ratios Table, we get  tan 60° = √3]

          = √3

R.H.S. = \(\frac{2 tan A}{1 - tan^{2} A}\)

          = \(\frac{2 tan 30°}{1 - tan^{2} 30°}\)

[From the Trigonometric Ratios Table, we get  tan 30° = \(\frac{√3}{3}\)]

          = \(\frac{2 × \frac{√3}{3}}{1 - (\frac{√3}{3})^{2}}\)

          = \(\frac{\frac{2√3}{3}}{1 - \frac{3}{9}}\)

          = \(\frac{\frac{2√3}{3}}{\frac{9}{9} - \frac{3}{9}}\)

          = \(\frac{\frac{2√3}{3}}{\frac{9 - 3}{9}}\)

          = \(\frac{\frac{2√3}{3}}{\frac{6}{9}}\)

          = \(\frac{\frac{2√3}{3}}{\frac{2}{3}}\)

          = \(\frac{2√3}{3}\) × \(\frac{3}{2}\)

          = \(\frac{6√3}{6}\)

          = √3

Therefore, L.H.S. = R.H.S.

4. Find the exact value of sin 45° cos 30° + cos 45° sin 30° using trigonometric table.

Solution:

sin 45° cos 30° + cos 45° sin 30°

Plug-in the exact value of sin 45°, cos 30°, cos 45° and sin 30° from the trig ratios table

= \(\frac{√2}{2}\) × \(\frac{√3}{2}\) + \(\frac{√2}{2}\) × \(\frac{1}{2}\)

= \(\frac{√6}{4}\) + \(\frac{√2}{4}\)

= \(\frac{√6 + √2}{4}\)

5. tan \(\frac{π}{3}\), tan \(\frac{π}{4}\), tan \(\frac{π}{6}\) are in geometric progression. 

Solution:

tan \(\frac{π}{3}\), tan \(\frac{π}{4}\), tan \(\frac{π}{6}\) are in geometric progression if,

tan\(^{2}\) \(\frac{π}{4}\) = tan \(\frac{π}{3}\) × tan \(\frac{π}{6}\)

Let's check 

L.H.S. = tan\(^{2}\) \(\frac{π}{4}\)

          = (1)², [From trig ratios table tan \(\frac{π}{4}\) = 1]

          = 1

R.H.S. = tan \(\frac{π}{3}\) × tan \(\frac{π}{6}\)

           = √3 × \(\frac{√3}{3}\)

           = \(\frac{3}{3}\)

           = 1

Therefore, tan\(^{2}\) \(\frac{π}{4}\) = tan \(\frac{π}{3}\) × tan \(\frac{π}{6}\)

Thus, tan \(\frac{π}{3}\), tan \(\frac{π}{4}\), tan \(\frac{π}{6}\) are in G.P. (geometric progression).

6. Find the value of \(\frac{4}{3}\) tan² 60° + 3 cos² 30° - 2 sec² 30° - \(\frac{3}{4}\) cot² 60° using trigonometric table.

Solution:

The given trigonometric expression is 

\(\frac{4}{3}\) tan² 60° + 3 cos² 30° - 2 sec² 30° - \(\frac{3}{4}\) cot² 60°

= \(\frac{4}{3}\) . (√3)²  + 3 . (\(\frac{√3}{2}\))² - 2 . (\(\frac{2}{√3}\))² - \(\frac{3}{4}\) . (\(\frac{1}{√3}\))²

[Since, the value of tan 60° = √3, cos 30° = \(\frac{√3}{2}\), sec 30° = \(\frac{2}{√3}\) and cot 60° = \(\frac{1}{√3}\)]

= \(\frac{4}{3}\) . 3 + 3 . \(\frac{3}{4}\) - 2 . \(\frac{4}{3}\) - \(\frac{3}{4}\) . \(\frac{1}{3}\)

= 4 + \(\frac{9}{4}\) - \(\frac{8}{3}\) - \(\frac{1}{4}\)

= \(\frac{10}{3}\)

= 3\(\frac{1}{3}\)

7. If A = 30°, show that cos 2A = cos² A - sin² A 

Solution:

L.H.S. = cos 2A

          = cos 2 . 30°

          = cos 60°

          = \(\frac{1}{2}\)

R.H.S. = cos² A - sin² A 

           = cos² 30° - sin² 30° 

          = (\(\frac{√3}{2}\))²  - (\(\frac{1}{2}\))²

          = \(\frac{3}{4}\) - \(\frac{1}{4}\)

          = \(\frac{1}{2}\)

Therefore, L.H.S. = R.H.S.

8.  If A = 30°, show that 3 sin A - 4 sin³ A = sin 3A

Solution:

L.H.S. = 3 sin A - 4 sin³ A

          = 3 sin 30° - 4 sin³ 30°

          = 3 . \(\frac{1}{2}\) - 4 (\(\frac{1}{2}\))³

          = \(\frac{3}{2}\) - 4 . \(\frac{1}{8}\)

          = \(\frac{3}{2}\) - \(\frac{1}{2}\)

          = 1

          = sin 90° (Using trigonometric table)

          = sin 3 . 30°

          = sin 3A = R.H.S.

Therefore, L.H.S. = R.H.S.

9. If the cosine of an angle complementary to θ be \(\frac{1}{2}\), find the value of cot θ.

Solution:

Angle complementary to θ is 90° - θ.

Therefore, by question, cos (90° - θ) = \(\frac{1}{2}\) =  cos 60°

Therefore, 90° - θ = 60°

            or, 90° - 60° = θ

            or, θ = 30°

Therefore, cot θ = cot 30° = √3. (Using trigonometric table)

10. Find a value of θ which satisfies the equation cos 3θ = sin 2θ.

Solution:

Since, sin 2θ = cos (90° - 2θ),

Therefore, from the given equation we get,

cos 3θ = sin 2θ = cos (90° - 2θ)

Therefore, 3θ = 90° - 2θ

           ⟹ 3θ + 2θ = 90° - 2θ + 2θ; [Adding 2θ on both sides]

           ⟹ 5θ = 90°

           ⟹ θ = \(\frac{90}{5}\)°

           ⟹ θ = 18°

Therefore, the required value of θ is 18°.

11. Each angle A, B, C of the triangle ABC is acute and sin (B + C - A) = 1, tan (C + A - B) =  √3; find A, B and C.

Solution:

Since A, B, C are the angles of ∆ABC

Therefore, A + B + C = 180°   ................... (i)

Again, sin (B + C - A) = 1 = sin 90°  (Using trigonometry table)

Therefore, B + C - A = 90°   ................... (ii)

[Since each of the angles A, B, C is acute, hence, B + C > A]

And tan (C + A - B) = √3 = tan 60°, (Using trigonometry table)

Therefore, C + A - B = 60°   ................... (iii)

[Since each of the angles A, B, C is acute, hence, C + A > B]

Now we add the equations (ii) and (iii)

We get 

(B + C - A) + (C + A - B) = 90° + 60°

⟹ (B + C - A + C + A - B) = 150°

⟹ 2C = 150°

⟹ C = \(\frac{150}{2}\)°

⟹ C = 75°

              

Again, we add the equations (i) and (ii)

(A + B + C) + (B + C - A) = 180° + 90°

⟹ (A + B + C + B + C - A) = 270°

⟹ 2B + 2C = 270°

⟹ 2B + 2(75°) = 270°; [Since C= 75°]

⟹ 2B + 150° = 270°

⟹ 2B = 270° - 150°

⟹ 2B = 120°

⟹ B = \(\frac{120}{2}\)°

⟹ B = 60°

Now, putting the values of B and C in (i) we get,

A + 60° + 75° = 180°

⟹ A + 135° = 180°

⟹ A = 180° - 135°

⟹ A = 45°

Therefore, the required angles are:

A = 45°;   B = 60°;   C = 75°

12.  If (x - 2)(sin \(\frac{π}{3}\) cos \(\frac{π}{6}\) + cos \(\frac{π}{3}\) sin \(\frac{π}{6}\)) = sec³ \(\frac{π}{3}\) + tan⁴ \(\frac{π}{4}\) - cosec² \(\frac{π}{6}\), find the value of x.

Solution:

(x - 2)(sin \(\frac{π}{3}\) cos \(\frac{π}{6}\) + cos \(\frac{π}{3}\) sin \(\frac{π}{6}\)) = sec³ \(\frac{π}{3}\) + tan⁴ \(\frac{π}{4}\) - cosec² \(\frac{π}{6}\)

⟹ (x - 2)(sin 60° cos 30° + cos 60° sin 30°) = sec³ 60° + tan⁴ 45° - cosec² 30°, [Since, π = 180°]

⟹ (x - 2)(\(\frac{√3}{2}\) . \(\frac{√3}{2}\) + \(\frac{1}{2}\) . \(\frac{1}{2}\)) = 2³ + 1⁴ - 2²

⟹ (x - 2)(\(\frac{3}{4}\) + \(\frac{1}{4}\)) = 8 + 1 - 4

⟹ (x - 2)(\(\frac{3 + 1}{4}\)) = 5

⟹ (x - 2)(\(\frac{4}{4}\)) = 5

⟹ (x - 2) (1) = 5

⟹ x - 2 = 5

⟹ x = 5 + 2

⟹ x = 7

Therefore, the value of x = 7.

Difference Between Trigonometric Table and Trigonometric Formula:

Trigonometric Table	Trigonometric Formula
Shows exact values	Shows identities
Used for calculations	Used for proofs
Limited to standard angles	Applied to general equations

Download Trigonometric Ratios Table PDF:

Trigonometric Ratios Table Quiz:

Trigonometric Ratios Quiz

Answer 10 randomly selected questions on standard-angle trigonometric values.

Question 1 of 10 Score: 0

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Quick Check Yourself: [🎯 Try these without looking at the table!]

1. What is sin 45° × cos 30°?

☐ \(\frac{\sqrt{6}}{4}\);    ☐ \(\frac{\sqrt{2}}{4}\);    ☐ \(\frac{1}{2}\)

Show Answer

sin 45° = 1/√2

cos 30° = √3/2

Multiply:

(1/√2) × (√3/2)

= √3/(2√2)

= (√3 × √2)/(2√2 × √2)

= √6/4

✅ Correct Answer: √6/4

2. What is the value of tan 60° – tan 30° ?

☐ \(\frac{2}{\sqrt{3}}\);    ☐ 1;    ☐ √3

Show Answer

tan 60° = √3

tan 30° = 1/√3

tan 60° - tan 30°

= √3 – 1/√3

= (3 – 1)/√3

= 2/√3

✅ Correct Answer: 2/√3

3. Which of the following is undefined?

☐ tan 90°;   ☐ sin 90°;   ☐ cos 0°

Show Answer

tan 90° is undefined.

sin 90° = 1

cos 90° = 0

Therefore, tan 90° = sin 90°/cos 90° = 1/0 = undefined

✅ Correct Answer: tan 90°

4. What is sec 60° × cos 60° ?

☐ 1;   ☐ 2;   ☐ \(\frac{1}{2}\)

Show Answer

sec 60° = 2

cos 60° = 1/2

Therefore, sec 60° × cos 60°

= 2 × 1/2 = 1

✅ Correct Answer: 1

5. Evaluate: (1 + tan² 45°) / (1 + sec² 30°)

☐ \(\frac{1}{3}\);   ☐ \(\frac{7}{6}\);   ☐ \(\frac{2}{3}\)

Show Answer

tan 45° = 1

Now, 1 + tan²45° = 1 + 1 = 2

Again, sec 30° = 2/√3

⟹ sec²30° = 4/3

⟹ 1 + sec²30° = 1 + 4/3 = 7/3

So, (1 + tan² 45°) / (1 + sec² 30°)

⟹ 2/(7/3)

⟹ 2 ÷ (7/3) ⟹ 6/7

✅ Correct Answer: 6/7

Worksheet on Trigonometrical Ratios Table:

Practice Questions:

1. If 30° then show that,

(i) 4 cos³ A - 3 cos A = cos 3A

(ii) \(\frac{1 - tan^{2} A}{1 + tan^{2} A}\) = cos 2A

2. If θ = 60° then prove that,

(i) cos θ = 2 cos² \(\frac{θ}{2}\) - 1 = cos² \(\frac{θ}{2}\) - sin² \(\frac{θ}{2}\)

(ii) \(\frac{2 tan \frac{θ}{2}}{1 + tan^{2}\frac{θ}{2}}\) = sin θ

3. Find the values of:

(i) sin 45° cos 30° + cos 45° sin 30°

(ii) cos² \(\frac{π}{4}\) - sin² \(\frac{π}{6}\) + tan² \(\frac{π}{3}\)

(iii) sin³ 30° + 4 cot³ 45° - cosec² 30°

(iv) cos \(\frac{π}{6}\) cot \(\frac{π}{3}\) + 2 . cos² \(\frac{π}{3}\)

Answer:

3. (i) \(\frac{√6 + √2}{4}\) or, \(\frac{√3 + 1}{2√2}\)

(ii) 3\(\frac{1}{4}\)

(iii) \(\frac{1}{8}\)

(iv) 1

4. Prove that,

(i) \(\frac{(sin  0° + sin  60°)(cos  60° + cot  45°)}{(cot  60° + tan  30°)(cosec  30° - cosec  90°)}\) = \(\frac{9}{8}\)

(ii) cos² 60°, cos² 45°, cos² 30° are in A.P.

(iii) tan² \(\frac{π}{3}\) - 2 tan² \(\frac{π}{4}\) = cot² \(\frac{π}{6}\) - 2 sin² \(\frac{π}{6}\) - \(\frac{3}{4}\) cosec² \(\frac{π}{4}\)

(iv) sec 60°, sec² 45°, sec² 30° are in H.P. (Harmonic Progression)

5. Find x if:

(i) (x + 1)cot² \(\frac{π}{6}\) = 2 cos² \(\frac{π}{3}\) + \(\frac{3}{4}\) sec² \(\frac{π}{4}\) + 4 sin² \(\frac{π}{6}\)

(ii) \(\frac{1 + 2 sin 60° cos 60°}{cos 60° + sin 60°}\) + \(\frac{1 - 2 sin 60° cos 60°}{sin 60° - cos 60°}\) = 2x

Answer:

5. (i) x = 0

(ii) x = \(\frac{√3}{2}\)

6. Solve (ϕ and β are positive acute angles): sin (ϕ - β) = \(\frac{1}{2}\); cos (ϕ + β)= \(\frac{1}{2}\) using trigonometry table.

Answer:

6. ϕ = 45° and β = 45°

7. The angle A of the triangle ABC is obtuse; if sec (B + C) cosec (B - C) = 2 find the angles.

Answer:

7. A = 120°, B = 45° and C = 15°

8. Solve ( 0 ≤ θ ≤ 90°)

(i) tan θ + cot θ = 2

(ii) 2 cos² θ + 5 sin θ = 4

(iii) tan ² θ - (√3 + 1) tan θ + √3 = 0

(iv) sec² θ + tan² θ = 7

(v) 2 sin² θ = 3(1 - cos θ)

(vi) 2 sin θ tan θ + 1 = tan θ + 2 sin θ

(vii) tan θ - cot θ = cosec θ.

Answer:

8. (i) θ = 45°

(ii) θ = 30°

(iii) θ = 45°, 60°

(iv) θ = 60°

(v) θ = 0°, 60°

(vi) θ = 30°, 45°

(vii) θ = 60°

9. Find a value of β for each of the following equations:

(i) sin 4β = cos β

(ii) tan 3β = cot β

(iii) sin 3β = cos 7β

Answer:

9. (i) β = 18°

(ii) β = 22.5°

(iii) β = 9°

10. α,  β, γ are positive acute angles and sin (α + β  - γ) = cos (β + γ - α) = tan (γ + α - β) = 1; find α, β and γ.

Answer:

10. α = 67.5°; β = 45°; γ = 22.5°

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About Author

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Last Updated: July 2026

Reviewed by: Math Only Math Editorial Team

Educational Level: Class 9–12

Subject: Mathematics

Topic: Trigonometry

● Trigonometric Functions

• Basic Trigonometric Ratios and Their Names
• Restrictions of Trigonometrical Ratios
• Reciprocal Relations of Trigonometric Ratios
• Quotient Relations of Trigonometric Ratios
  
• Limit of Trigonometric Ratios
  
• Trigonometrical Identity
• Problems on Trigonometric Identities
• Elimination of Trigonometric Ratios
• Eliminate Theta between the equations
• Problems on Eliminate Theta
• Trig Ratio Problems
• Proving Trigonometric Ratios
• Trig Ratios Proving Problems
• Verify Trigonometric Identities
• Trigonometrical Ratios of 0°
• Trigonometrical Ratios of 30°
• Trigonometrical Ratios of 45°
• Trigonometrical Ratios of 60°
• Trigonometrical Ratios of 90°
• Trigonometrical Ratios Table
• Problems on Trigonometric Ratio of Standard Angle
  
• Trigonometrical Ratios of Complementary Angles
• Rules of Trigonometric Signs
• Signs of Trigonometrical Ratios
  
• All Sin Tan Cos Rule
• Trigonometrical Ratios of (- θ)
• Trigonometrical Ratios of (90° + θ)
• Trigonometrical Ratios of (90° - θ)
• Trigonometrical Ratios of (180° + θ)
• Trigonometrical Ratios of (180° - θ)
• Trigonometrical Ratios of (270° + θ)
• Trigonometrical Ratios of (270° - θ)
• Trigonometrical Ratios of (360° + θ)
• Trigonometrical Ratios of (360° - θ)
• Trigonometrical Ratios of any Angle
• Trigonometrical Ratios of some Particular Angles
• Trigonometric Ratios of an Angle
• Trigonometric Functions of any Angles
• Problems on Trigonometric Ratios of an Angle
• Problems on Signs of Trigonometrical Ratios

11 and 12 Grade Math

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