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

Problem 1 :

Evaluate tan 765°.

Solution : 

The given 765° is greater than 360°.

So, we have to divide 765 by 360 and take the remainder. 

When 765 is divided by 360, the remainder is 15. 

Therefore,

765°  =  45°

Then,

tan 765°  =  tan 45°

tan 765°  =  1

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°

Then,

cos 870°  =  cos 150° 

cos 870°  =  cos (180° - 30°)

cos 870°  =  - cos 30°

cos 870  =  - √3/2

ASTC Formula

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.

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. 

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

Evaluation of Trigonometric Ratios using 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. 

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.

Finding the value of sin 780° :

sin 780°  =  sin (2 ⋅ 360° + 60°)

sin 780°  =  sin 60°

sin 780°  =   √3/2

Finding the value of sin 480° :

sin 480°  =  sin (360° + 120°)

sin 480°  =  sin 120°

sin 480°  =  sin (180° - 60°)

sin 480°  =  sin 60°

sin 480°  =  √3/2

Finding the value of sin 120° : 

cos 120°  =  cos (180° - 60°)

cos 120°  =  - cos 60°

cos 120°  =  - 1/2  

The value of cos 60° is 1/2.    

sin 780° sin 480° + cos 120° cos 60° : 

=  (√3/2) ⋅ (√3/2)  +  (-1/2) ⋅ (1/2)

=  (3/4) - (1/4)

=  (3 - 1) / 4

=  2/4

=  1/2

So, the value of the given trigonometric expression is 1/2. 

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

Then, the given trigonometric expression is equal to 

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

=  1 

So, the simplified form of the given trigonometric expression is equal to 1.   

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  

Then the given expression is equal to 

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

=  - sec θ ⋅ tan θ  +  sec θ ⋅ tan θ

=  0  

So, the simplified of the given trigonometric expression is equal to 0.    

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