TRIGONOMETRIC FUNCTIONS OF ANGLES GREATER THAN 360 DEGREES

About "Trigonometric functions of angles greater than 360 degrees"

Trigonometric functions of angles greater than 360 degrees :

In the trigonometric functions sin θ, cos θ, tan θ, csc θ, sec θ and cot θ, if the angle θ is greater than or equal to 360°, we have to do the following steps.

(i)  Divide the given angle by 360° and

(ii)  Take the remainder

For example,

For example,  

(i) Let us consider the angle 450°.

When we divide 450° by 360, we get the remainder 90°. 

Therefore, 450°  =  90°

(ii) Let us consider the angle 360°

When we divide 360° by 360, we get the remainder 0°.

Therefore, 360°  =  0°

More clearly, 

450°  =  90° means, after having completed one circle of 360°, the position of the angle will be at 90°

It has been given in the figure given below. 

Trigonometric functions of angles greater than 360 degrees - Practice problems

Let us look at some practice problems on "Trigonometric functions of angles greater than 360 degrees"

Problem 1 :

Evaluate : tan 735°

Solution : 

The given 735° is greater than 360°.

So, we have to divide 735° by 360 and take the remainder. 

When 735° is divided by 360, the remainder is 15°. 

Therefore,

735°  =  15° ------> tan 735°  =  tan 15°

Hence, tan 735° is equal to tan 15°   

Let us look at the next problem on "Trigonometric functions of angles greater than 360 degrees"

Problem 2 :

Evaluate : cos (-870°)

Solution : 

Since the given angle (-870°) has negative sign, we have to assume it falls in the fourth quadrant.

In the fourth quadrant, "cos" is positive. 

So, we have cos (-870°)  =  cos 870°.

The given 870° is greater than 360°.

So, we have to divide 870° by 360 and take the remainder. 

When 870° is divided by 360, the remainder is 150°. 

Therefore,

870°  =  150° ------> cos 870°  =  cos 150° 

cos 870°  =  cos (180° - 30°)

cos 870°  =  - cos 30°

cos 870  =  - √3 / 2

Hence, cos 870° is equal to √3 / 2 .

In the above problem 2, we have cos (180° - 30°) and it has been evaluated as " - cos 30 ".

If we want to know "How it has been evaluated", first, we have to understand ASTC formula.

The ASTC formula can be remembered easily using the following phrases.

"All Sliver Tea Cups" 

or

"All Students Take Calculus"

ASTC formula has been explained clearly in the figure given below.

More clearly 

In the first quadrant (0° to 90°), all trigonometric ratios are positive.

In the second quadrant (90° to 180°), sin and csc are positive and other trigonometric ratios are negative.

In the third quadrant (180° to 270°), tan and cot are positive and other trigonometric ratios are negative.

In the fourth quadrant (270° to 360°), cos and sec are positive and other trigonometric ratios are negative.

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Important conversions

When we have the angles 90° and 270° in the trigonometric ratios in the form of

(90° + θ)

(90° - θ)

(270° + θ)

(270° - θ)

We have to do the following conversions, 

sin θ <------> cos θ

tan θ <------> cot θ

csc θ <------> sec θ

For example,

sin (270° + θ)  =  - cos θ

cos (90° - θ)  =  sin θ

For the angles 0° or 360° and  180°, we should not make the above conversions. 

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Division of Quadrants 

(90° - θ) -------> I st Quadrant

(90° + θ) and (180° - θ) -------> II nd Quadrant

(180° + θ) and (270° - θ) -------> III rd Quadrant

(270° + θ), (360° - θ) and (θ) -------> IV th Quadrant

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Evaluation of trigonometric ratios using ASTC formula

Let us see, how to use ASTC formula.

Example 1 :

Evaluate : cos (270° - θ)

Solution :

To evaluate cos (270° - θ), we have to consider the following important points. 

(i)  (270° - θ) will fall in the III rd quadrant. 

(ii)  When we have 270°, "cos" will become "sin"

(iii)  In the III rd quadrant, the sign of "cos" is negative. 

Considering the above points, we have 

cos (270° - θ)  =  - sin θ

Example 2 :

Evaluate : sin (180° + θ)

Solution :

To evaluate sin (180° + θ), we have to consider the following important points. 

(i)  (180° + θ) will fall in the III rd quadrant. 

(ii)  When we have 180°, "sin" will not be changed

(iii)  In the III rd quadrant, the sign of "sin" is negative. 

Considering the above points, we have 

sin (180° + θ)  =  - sin θ

Based on the above two examples, we can evaluate the following trigonometric ratios. 

sin (-θ)  =  - sin θ

cos (-θ)  =  cos θ

tan (-θ)  =  - tan θ

csc (-θ)  =  - csc θ

sec (-θ)  =  sec θ

cot (-θ)  =  - cot θ

sin (90°-θ)  =  cos θ

cos (90°-θ)  =  sin θ

tan (90°-θ)  =  cot θ

csc (90°-θ)  =  sec θ

sec (90°-θ)  =  csc θ

cot (90°-θ)  =  tan θ

sin (90°+θ)  =  cos θ

cos (90°+θ)  =  -sin θ

tan (90°+θ)  =  -cot θ

csc (90°+θ)  =  sec θ

sec (90°+θ)  =  -csc θ

cot (90°+θ)  =  -tan θ

sin (180°-θ)  =  sin θ

cos (180°-θ)  =  -cos θ

tan (180°-θ)  =  -tan θ

csc (180°-θ)  =  csc θ

sec (180°-θ)  =  -sec θ

cot (180°-θ)  =  -cot θ

sin (180°+θ)  =  -sin θ

cos (180°+θ)  =  -cos θ

tan (180°+θ)  =  tan θ

csc (180°+θ)  =  -csc θ

sec (180°+θ)  =  -sec θ

csc (180°+θ)  =  cot θ

sin (270°-θ)  =  -cos θ

cos (270°-θ)  =  -sin θ

tan (270°-θ)  =  cot θ

csc (270°-θ)  =  -sec θ

sec (270°-θ)  =  -csc θ

cot (270°-θ)  =  tan θ

sin (270°+θ)  =  -cos θ

cos (270°+θ)  =  sin θ

tan (270°+θ)  =  -cot θ

csc (270°+θ)  =  -sec θ

sec (270°+θ)  =  cos θ

cot (270°+θ)  =  -tan θ

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Angles greater than or equal to 360°

If the angle is equal to or greater than 360°, we have to divide the given angle by 360 and take the remainder. 

For example,  

(i) Let us consider the angle 450°.

When we divide 450° by 360, we get the remainder 90°. 

Therefore, 450°  =  90°

(ii) Let us consider the angle 360°

When we divide 360° by 360, we get the remainder 0°.

Therefore, 360°  =  0°

Based on the above two examples, we can evaluate the following trigonometric ratios. 

sin (360° - θ)  =  sin (0° - θ)  =  sin (θ)  =  - sin θ

cos (360° - θ)  =  cos (0° - θ)  =  cos (θ)  =  cos θ

tan (360° - θ)  =  tan (0° - θ)  =  tan (θ)  =  - tan θ

csc (360° - θ)  =  csc (0° - θ)  =  csc (θ)  =  - csc θ

sec (360° - θ)  =  sec (0° - θ)  =  sec (θ)  =  sec θ

cot (360° - θ)  =  cot (0° - θ)  =  cot (θ)  =  - cot θ

sin (360° + θ)  =  sin (0° + θ)  =  sin θ

cos (360° + θ)  =  cos (0° + θ)  =  cos θ

tan (360° + θ)  =  tan (0° + θ)  =  tan θ

csc (360° + θ)  =  csc (0° + θ)  =  csc θ

sec (360° + θ)  =  sec (0° + θ)  =  sec θ

cot (360° + θ)  =  cot (0° + θ)  =  cot θ

ASTC formula - Practice problems

Problem 1 :

Find the value of  (sin 780 sin 480° + cos 120° cos 60°)

Solution : 

Let us find the value of each trigonometric ratio for the given angle.

sin 780°  =  sin 60°  =   √3 / 2

sin 480°  =  sin 120°  =  sin (180° - 60°)  =  sin 60°  =  √3 / 2

cos 120°  =  cos (180° - 60°)  =  - cos 60°  =  - 1 / 2  

cos 60°  =  1/2   

sin780 sin480° + cos120° cos60°  =  (√3/2) x (√3/2)  +  (-1/2) x (1/2)

=  (3/4) - (1/4)

=  (3-1) / 4

=  2 / 4

=  1/2

Hence, the value of the given trigonometric expression is equal to 1/2. 

Let us look at the next problem on "Trigonometric functions of angles greater than 360 degrees"

Problem 2 :

Simplify :

cot (90°-θ) sin (180°+θ) sec(360°-θ) / tan(180°+θ) sec(-θ) cos(90°+θ)

Solution : 

Using ASTC formula, we have

cot (90°-θ)   =  tan θ

sin (180°+θ)  =  - sin θ

sec(360°-θ)  =  sec θ

tan(180°+θ)  =  tan θ  

sec(-θ)  =  sec θ  

cos(90°+θ)  =  - sin  θ

The given expression  is

= (tan θ x -sinθ x sec θ) / (tan θ x sec θ x -sin θ)

= 1 

Hence, the simplification of the given trigonometric expression is equal to 1.    

Let us look at the next problem on "Trigonometric functions of angles greater than 360 degrees"

Problem 3 :

Simplify :

sec(360°-θ) tan(180°-θ)  + cot(90°+θ) co(270°-θ)

Solution : 

Using ASTC formula, we have

sec (360°-θ)   =  sec θ

tan (180°-θ)  =  - tan θ

cot (90°+θ)  =  - tan θ

  cos (270°-θ)  =  - sec θ  

The given expression  is

=  sec θ x (-tan θ)  +  (-tan θ) x (-sec θ)

=  - sec θ x tan θ  +  sec θ x tan θ

=  0  

Hence, the simplification of the given trigonometric expression is equal to 0.    

After having gone through the stuff given above, we hope that the students would have understood "Trigonometric functions of angles greater than 360 degrees"

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