ML Aggarwal Class 10 Maths Solutions Chapter 19 Trigonometric Tables

Exercise 19

Question 1. Find the value of the following:
(i) sin 35° 22'
(ii) sin 71° 31'
(iii) sin 65° 20'
(iv) sin 23° 56'
Answer:
(i) sin 35° 22' = sin (35° 18' + 4')
sin 35° 18' = .5779
Mean difference for 4' = .0010 (To be added)
sin 35° 22' = .5779 + .0010 = .5789
Therefore, sin 35° 22' = .5789

(ii) sin 71° 31' = sin (71° 30' + 1')
sin 71° 30' = .9483
Mean difference for 1' = .0001 (To be added)
sin 71° 31' = .9483 + .0001 = .9484
Therefore, sin 71° 31' = .9484

(iii) sin 65° 20' = sin (65° 18' + 2')
sin 65° 18' = .9085
Mean difference for 2' = .0002 (To be added)
sin 65° 20' = .9085 + .0002 = .9087
Therefore, sin 65° 20' = .9087

(iv) sin 23° 56' = sin (23° 54' + 2')
sin 23° 54' = 0.4051
Mean difference of 2' = .0005 (To be added)
sin 23° 56' = .4051 + .0005 = .4056
Therefore, sin 23° 56' = .4056
In simple words: To find the sine of an angle between table entries, first locate the nearest table value. Then find the difference between your desired value and the table entry. Multiply this difference by the mean difference per minute, then add or subtract from the base value depending on whether you are going up or down in the tables.

Exam Tip: Always note whether you are adding or subtracting the mean difference - sine increases with angle, so you add when the angle is greater than the table entry.

Question 2. Find the value of the following:
(i) cos 62° 27'
(ii) cos 3° 11'
(iii) cos 86° 40'
(iv) cos 45° 58'
Answer:
(i) cos 62° 27' = cos (62° 24' + 3')
cos 62° 24' = 0.4633
Mean difference of 3' = .0008 (To be subtracted)
cos 62° 27' = .4633 - .0008 = .4625
Therefore, cos 62° 27' = .4625

(ii) cos 3° 11' = cos (3° 6' + 5')
cos 3° 6' = .9985
Mean difference of 5' = .0001 (To be subtracted)
cos 3° 11' = .9985 - .0001 = .9984
Therefore, cos 3° 11' = .9984

(iii) cos 86° 40' = cos (86° 36' + 4')
cos 86° 36' = .0593
Mean difference of 4' = .0012 (To be subtracted)
cos 86° 40' = .0593 - .0012 = .0581
Therefore, cos 86° 40' = .0581

(iv) cos 45° 58' = cos (45° 54' + 4')
cos 45° 54' = .6959
Mean difference of 4' = .0008 (To be subtracted)
cos 45° 58' = .6959 - .0008 = .6951
Therefore, cos 45° 58' = .6951
In simple words: Cosine values drop as angles increase. When finding a value between table entries, subtract the mean difference from the base table value, not add it.

Exam Tip: Remember the key difference - sine increases with angle (add the mean difference), while cosine decreases with angle (subtract the mean difference). Mixing these up is a common mistake.

Question 3. Find the value of the following:
(i) tan 15° 2'
(ii) tan 53° 14'
(iii) tan 82° 18'
(iv) tan 6° 9'
Answer:
(i) tan 15° 2' = tan (15° 0' + 2')
tan 15° 0' = .2679
Mean difference of 2' = .0006 (To be added)
tan 15° 2' = .2679 + .0006 = .2685
Therefore, tan 15° 2' = .2685

(ii) tan 53° 14' = tan (53° 12' + 2')
tan 53° 12' = 1.3367
Mean difference of 2' = .0016 (To be added)
tan 53° 12' = 1.3367 + .0016 = 1.3383
Therefore, tan 53° 14' = 1.3383

(iii) tan 82° 18' = 7.3962 (directly from tables)
Therefore, tan 82° 18' = 7.3962

(iv) tan 6° 9' = tan (6° 6' + 3')
tan 6° 6' = .1069
Mean difference of 3' = .0009 (To be added)
tan 6° 9' = .1069 + .0009 = .1078
Therefore, tan 6° 9' = 0.1078
In simple words: Tangent values get larger as angles get larger. Look up the closest angle in the table, then add the mean difference to get your answer.

Exam Tip: Some angles like 82° 18' already appear in the table - when this happens, read the value straight from the table without doing any mean difference calculation.

Question 4. Use tables to find the acute angle θ, given that
(i) sin θ = .5789
(ii) sin θ = .9484
(iii) sin θ = .2357
(iv) sin θ = .6371
Answer:
(i) Given: sin θ = .5789
sin 35° 18' = .5779 (From tables)
Difference = .0010
Mean difference for 4' = .0010
θ = 35° 18' + 4' = 35° 22'
Therefore, θ = 35° 22'

(ii) Given: sin θ = .9484
sin 71° 30' = .9483 (From tables)
Difference = .0001
Mean difference for 1' = .0001
θ = 71° 30' + 1' = 71° 31'
Therefore, θ = 71° 31'

(iii) Given: sin θ = .2357
sin 13° 36' = .2351 (From tables)
Difference = .0006
Mean difference for 2' = .0006
θ = 13° 36' + 2' = 13° 38'
Therefore, θ = 13° 38'

(iv) Given: sin θ = .6371
sin 39° 30' = .6361 (From tables)
Difference = .0010
Mean difference for 4' = .0009 and mean difference for 5' = .0011
Taking mean difference for 4' = .0009
θ = 39° 30' + 4' = 39° 34'
Therefore, θ = 39° 34'
In simple words: When you know the sine value and need to find the angle, look for the closest sine value in the table. Find how much your value differs from it, then work backwards using the mean difference to get the angle in degrees and minutes.

Exam Tip: When the difference falls between two mean difference values (as in part iv), pick the closer one - the calculated angle will be sufficiently accurate for exam purposes.

Question 5. Use tables to find the acute angle θ, given that
(i) cos θ = .4625
(ii) cos θ = .9906
(iii) cos θ = .6951
(iv) cos θ = .3412
Answer:
(i) Given: cos θ = .4625
cos 62° 30' = .4617 (From tables)
Difference = .0008
Mean difference for 3' = .0008
θ = 62° 30' - 3' = 62° 27'
Therefore, θ = 62° 27'

(ii) Given: cos θ = .9906
cos 7° 54' = .9905 (From tables)
Difference = .0001
Mean difference for 3' = .0001
θ = 7° 54' - 3' = 7° 51'
Therefore, θ = 7° 51'

(iii) Given: cos θ = .6951
cos 46° 0' = .6947 (From tables)
Difference = .0004
Mean difference for 2' = .0004
θ = 46° 0' - 2' = 45° 58'
Therefore, θ = 45° 58'

(iv) Given: cos θ = .3412
cos 70° 6' = .3404 (From tables)
Difference = .0008
Mean difference for 3' = .0008
θ = 70° 6' - 3' = 70° 3'
Therefore, θ = 70° 3'
In simple words: For cosine, find the nearest table value and check your given number against it. Since cosine drops as angle grows, subtract the mean difference minutes from the table angle to get your final angle.

Exam Tip: The key rule is - for cosine you always subtract the mean difference minutes, never add them. This is opposite to sine and tangent.

Question 6. Use tables to find the acute angle θ, given that
(i) tan θ = .2685
(ii) tan θ = 1.7451
(iii) tan θ = 3.1749
(iv) tan θ = .9347
Answer:
(i) Given: tan θ = .2685
tan 15° 0' = .2679 (From tables)
Difference = .0006
Mean difference for 2' = .0006
θ = 15° 0' + 2' = 15° 2'
Therefore, θ = 15° 2'

(ii) Given: tan θ = 1.7451
tan 60° 6' = 1.7391 (From tables)
Difference = .0060
Mean difference for 5' = .0060
θ = 60° 6' + 5' = 60° 11'
Therefore, θ = 60° 11'

(iii) Given: tan θ = 3.1749
tan 72° 30' = 3.1716 (From tables)
Difference = .0033
Mean difference for 1' = .0033
θ = 72° 30' + 1' = 72° 31'
Therefore, θ = 72° 31'

(iv) Given: tan θ = .9347
tan 43° 0' = .9325 (From tables)
Difference = .0022
Mean difference for 4' = .0022
θ = 43° 0' + 4' = 43° 4'
Therefore, θ = 43° 4'
In simple words: Search for your tangent value in the table, find the closest match, then determine the extra minutes needed by using the mean difference. Add those minutes to the table angle.

Exam Tip: Tangent values increase with angle, so always add the mean difference minutes to the table angle - never subtract.

Question 7. Using trigonometric table, find the measure of the angle A when sin A = 0.1822.
Answer: sin A = 0.1822
sin 10° 30' = 0.1822 (From tables)
A = 10° 30'
Therefore, the angle A = 10° 30'
In simple words: The sine value 0.1822 appears exactly in the table at 10° 30'. No mean difference calculation is needed - just read the angle straight from the table.

Exam Tip: When a value matches a table entry exactly, you do not need to apply mean differences - simply state the corresponding angle.

Question 8. Using tables, find the value of 2 sin θ - cos θ when
(i) θ = 35°
(ii) tan θ = .2679
Answer:
(i) Substituting θ = 35° into 2 sin θ - cos θ:
=\> 2 sin 35° - cos 35°
=\> 2 × .5736 - .8192
=\> 1.1472 - .8192
=\> .3280
Therefore, 2 sin θ - cos θ = .3280

(ii) Given: tan θ = .2679
tan θ = tan 15°
θ = 15°
Substituting θ = 15° into 2 sin θ - cos θ:
=\> 2 sin 15° - cos 15°
=\> 2 × .2588 - .9659
=\> .5176 - .9659
=\> -.4483
Therefore, 2 sin θ - cos θ = -0.4483
In simple words: Find the angle θ using tables, then look up the sine and cosine values for that angle. Plug these values into the given expression and solve using basic arithmetic.

Exam Tip: Always verify your angle is correct before computing - a mistake in finding θ will make your entire final answer wrong.

Question 9. If sin x° = 0.67, find the value of
(i) cos x°
(ii) cos x° + tan x°
Answer:
(i) Given: sin x° = 0.67
sin 42° = .6691 (From tables)
Difference = .0009
Mean difference for 4' = .0009
x° = 42° + 4' = 42° 4'
Finding cos x°:
cos 42° = .7431
Mean difference of 4' = .0008
cos 42° 4' = .7431 - .0008 = .7423
Therefore, cos 42° 4' = .7423

(ii) Calculating tan x°:
tan 42° = .9004
Mean difference of 4' = .0021
tan 42° 4' = .9004 + .0021 = .9025
=\> cos x° + tan x° = .7423 + .9025 = 1.6448
Therefore, cos x° + tan x° = 1.6448
In simple words: First use the given sine to find the angle x. Then look up both cosine and tangent values for that angle. Finally, add them together to get your answer.

Exam Tip: Make sure you apply the mean differences correctly for each trig function - cosine gets smaller so subtract, while sine and tangent get larger so add.

Question 10. If θ is acute and cos θ = .7258, find the value of
(i) θ
(ii) 2 tan θ - sin θ
Answer:
(i) Given: cos θ = .7258
cos 43° 30' = .7254
Difference = .0004
Mean difference of 2' = .0004
θ = 43° 30' - 2' = 43° 28'
Therefore, θ = 43° 28'

(ii) Substituting θ = 43° 28' into 2 tan θ - sin θ:
Finding value of tan 43° 28':
tan 43° 24' = .9457
Mean difference of 4' = .0022
tan 43° 28' = .9457 + .0022 = .9479
Finding value of sin 43° 28':
sin 43° 24' = .6871
Mean difference of 4' = .0008
sin 43° 28' = .6871 + .0008 = .6879
Calculating 2 tan θ - sin θ:
=\> 2 × .9479 - .6879
=\> 1.8958 - .6879
=\> 1.2079
Therefore, 2 tan θ - sin θ = 1.2079
In simple words: Start by finding the angle using the cosine value. Then look up the sine and tangent for that angle in the tables. Finally, substitute these values into the expression 2 tan θ - sin θ and calculate.

Exam Tip: When finding intermediate angles using mean differences, carry the exact angle (with minutes) through all subsequent calculations - rounding early will introduce errors in your final answer.