Database - Trigonometry

Ahm97sic SG

Question

Find all the angles between 0 degree and 360 degrees which satisfy the equation

sin (x + 63.434) = 0.632456

Answer

Basic angle (or Reference angle), theta

= sin-1 (0.632456)

= 39.232 degrees

Since sin +, so it lies in the 1st and 2nd quadrant,

x + 63.434 = 39.232, 180 - 39.232

x + 63.434 = 39.232, 140.768

x = 39.232 - 63.434, 140.768 - 63.434

x = - 24.2 , 77.33

Since x = - 24.2 degrees, it is a negative angle, we will need to convert it into a

positive angle 360 - 24.2 = 335.8 degrees.

Since x = 77.33 degrees and x = 335.8 degrees are between 0 degree and 360,

degrees. So, the answers are x = 77.33 degrees and x = 335.8 degrees

Check Answer :

(a) Substitute x = 77.33 degrees into

sin (x + 63.434) = sin (77.33 + 63.434) = 0.632456 (Correct)

(b) Substitute x = 335.8 degrees into

sin (x + 63.434) = sin (335.8 + 63.434) = 0.632456 (Correct)

Ahm97sic SG

Question

Without using a calculator,

evaluate tan ( tan-1 2 + tan-1 8)

(The number in red is in superscript)

Answer

tan (tan-1 2 + tan-1 8)

Let tan-1 2 = x. Hence, tan x = 2

Let tan-1 8 = y. Hence, tan y = 8

tan (x + y)

= [tan x + tany] / [1 - tanx tany]

= [2 + 8 ] / [1 - (2)(8)]

= - 2/3

Garrick_3658 SG

I think you forgot:

Let tan-1 8 = y. Hence, tan y = 8

Thanks.

Garrick_3658 SG

Can we write it as

Since 0<x<360, 63.434<x+63.434<423.434

And so we take it to go another round,

Basic angle (alpha) = 39.232

x + 63.434 = 140.768, 39.232+360

Is it considered a mistake in the exam?

Ahm97sic SG

Dear Darrick_3658,

Yes, I have forgotten to type

Let tan-1 8 = y. Hence, tan y = 8

Thanks.

Regards,

ahm9s7ic

Ahm97sic SG

Dear Darrick_3658,

There are two approaches to solve this question.

The first approach is the typical approach that I used to solve the question. However, one of the answers gives a negative angle and hence we need to convert it into a positive angle and check whether this positive angle still falls within the required range. If it still falls within the required range, it will be included as one of the answers.

The second approach is the approach that is given in your answer. It is ok as the final answer x + 63.434 = 39.232 + 360

x = 335.768

and it is still within the required range for x ie between 0 and 360 and hence it will be accepted as one of the answers.

However, this second approach avoids the understanding of negative angles.

There are questions that asked students to find all the angles between - 180 degrees and 180 degrees. For example, Shing Lee Add Maths Textbook Page 173 Exercise 6D Question 6(a) to 6(f).

Question 6(f) [Shing Lee Add Maths Textbook]

Find all the angles, x, where - 180 degrees < x < 180 degrees such that

5 tan (x + 10 degrees) = - 6

Another question that involves negative angle

Question 7(d) [Pan Pacific Add Maths Textbook]

Find all the angles, x, where 0 degree < x < 360 degrees such that

2 sin (-x) = 0.3

Dear Darrick_3658, thank you very much for your help so that my careless mistakes can be corrected. I appreciate it very much.

Thank you very much for your help, participation and support. Please come more often.

Warmest Regards,

ahm97sic

iamapebble SG

YAY GOT THE ANSWER! :D Hmmm, if they say the range of angles is between 0 to 180, instead of adding 360, we add 180 right?

Ahm97sic SG

YAY GOT THE ANSWER! :D Hmmm, if they say the range of angles is between 0 to 180, instead of adding 360, we add 180 right?

Dear iamapebble,

If the range of the angles is between 0 degree and 180 degrees, we still do the same workings except that when we get the answers, we will check which of the answers are within the range of between 0 degree and 180 degrees. Any answers that fall within the range of between 0 degree and 180 degrees will be accepted as the answers.

In addition, if one of the answers gives a negative angle and hence we need to convert it into a positive angle and check whether this positive angle still falls within the required range. If it still falls within the required range, it will be included as one of the answers.

Regards,

ahm97sic

iamapebble SG

Ohhh I see I see, so for the negative one for 0 to 180, by adding 180 is correct right? ^^

Ahm97sic SG

Ohhh I see I see, so for the negative one for 0 to 180, by adding 180 is correct right? ^^

Dear iamapebble,

If the range of the angles is between - 180 degrees and 180 degrees, we still do the same workings except that when we get the answers, we will check which of the answers are within the range of between -180 degrees and 180 degrees. Any answers that fall within the range of between -180 degree and 180 degrees will be accepted as the answers.

This means that we will check both the negative angle(s) and positive angle(s) are within the range of between -180 degrees and 180 degrees. Any answers that fall within the range of between -180 degree and 180 degrees will be accepted as the answers.

It is to be noted that if one of the answers is 226 degrees, we need to convert it into a negative angle ie 226 - 360 = - 134 degrees. Since it is within the range of between -180 degrees and 180 degrees , -134 degrees will be accepted as one of the answers.

Similarly, it is to be noted that if one of the answers is - 281 degrees, we need to convert it into a positive angle ie -281 + 360 = 79 degrees. Since it is within the range of between -180 degrees and 180 degrees , 79 degrees will be accepted as one of the answers.

Regards,

ahm97sic

iamapebble SG

OHH okok I get it, thank youuu :D

Ahm97sic SG

Dear iamapebble,

You are welcome.

Please come into my forums (addmaths.sgforums.com ie add maths forums and emaths.sgforums.com ie emaths forum) more often. OK.

Regards,

ahm97sic

Ahm97sic SG

For "O" Level Add Maths students to try,

Question

Solve cos x + (sq root 2) cos 3x + sin x = 0 for x between 0 and pi.