TRIGONOMETRIC RATIOS OF NEGATIVE ANGLES
About "Trigonometric ratios of negative angles"
"Trigonometric ratios of negative angles" is one of the branches of ASTC formula in trigonometry.
Trigonometric-ratios of negative angles are given below.
sin (- θ) = - sin θ
cos (- θ) = cos θ
tan (- θ) = - tan θ
csc (- θ) = -csc θ
sec (- θ) = sec θ
cot (- θ) = - cot θ
Let us see, how the trigonometric ratios of negative angles are determined.
To know that, 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 formla has been explained clearly in the figure given below.
More clearly
Key Concept - Negative angles
Whenever we have negative angles in trigonometric ratios, we have to assume that it falls in the fourth quadrant.
In the fourth quadrant ( - θ or 360° - θ ), 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.
Evaluation of trigonometric ratios using ASTC formula
Problem 1 :
Evaluate : sin (- θ)
Solution :
sin (- θ) can be written as sin (0° - θ)
That is, sin (- θ) = sin (0° - θ)
To evaluate sin (0° - θ), we have to consider the following important points.
(i) (0° - θ) will fall in the IV th quadrant.
(ii) When we have 0°, "sin" will not be changed as "cos"
(iii) In the IV th quadrant, the sign of "sin" is negative.
Considering the above points, we have
sin (- θ) = sin (0° - θ) = - sin θ
Let us look at the next stuff on "Trigonometric ratios of negative angles"
Problem 2 :
Evaluate : cos (- θ)
Solution :
cos (- θ) can be written as cos (0° - θ)
That is, cos (- θ) = cos (0° - θ)
To evaluate cos (0° - θ), we have to consider the following important points.
(i) (0° - θ) will fall in the IV th quadrant.
(ii) When we have 0°, "cos" will not be changed as "sin"
(iii) In the IV th quadrant, the sign of "cos" is positive.
Considering the above points, we have
cos (- θ) = cos (0° - θ) = cos θ
Let us look at the next stuff on "Trigonometric ratios of negative angles"
Problem 3 :
Evaluate : tan (- θ)
Solution :
tan (- θ) can be written as tan (0° - θ)
That is, tan (- θ) = tan (0° - θ)
To evaluate tan (0° - θ), we have to consider the following important points.
(i) (0° - θ) will fall in the IV th quadrant.
(ii) When we have 0°, "tan" will not be changed as "cot"
(iii) In the IV th quadrant, the sign of "tan" is negative.
Considering the above points, we have
tan (- θ) = tan (0° - θ) = - tan θ
Let us look at the next stuff on "Trigonometric ratios of negative angles"
Problem 4 :
Evaluate : csc (- θ)
Solution :
csc (- θ) can be written as csc (0° - θ)
That is, csc (- θ) = csc (0° - θ)
To evaluate csc (0° - θ), we have to consider the following important points.
(i) (0° - θ) will fall in the IV th quadrant.
(ii) When we have 0°, "csc" will not be changed as "sec"
(iii) In the IV th quadrant, the sign of "csc" is negative.
Considering the above points, we have
csc (- θ) = csc (0° - θ) = - csc θ
Let us look at the next stuff on "Trigonometric ratios of negative angles"
Problem 5 :
Evaluate : sec (- θ)
Solution :
sec (- θ) can be written as sec (0° - θ)
That is, sec (- θ) = sec (0° - θ)
To evaluate sec (0° - θ), we have to consider the following important points.
(i) (0° - θ) will fall in the IV th quadrant.
(ii) When we have 0°, "sec" will not be changed as "csc"
(iii) In the IV th quadrant, the sign of "sec" is positive.
Considering the above points, we have
sec (- θ) = sec (0° - θ) = sec θ
Let us look at the next stuff on "Trigonometric ratios of negative angles"
Problem 6 :
Evaluate : cot (- θ)
Solution :
cot (- θ) can be written as cot (0° - θ)
That is, cot (- θ) = cot (0° - θ)
To evaluate cot (0° - θ), we have to consider the following important points.
(i) (0° - θ) will fall in the IV th quadrant.
(ii) When we have 0°, "cot" will not be changed as "tan"
(iii) In the IV th quadrant, the sign of "cot" is negative.
Considering the above points, we have
cot (- θ) = cot (0° - θ) = - cot θ
Summary (Negative angles)
sin (- θ) = - sin θ
cos (- θ) = cos θ
tan (- θ) = - tan θ
csc (- θ) = -csc θ
sec (- θ) = sec θ
cot (- θ) = - cot θ
Angles 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
Click here to know more about ASTC formula
After having gone through the stuff given above, we hope that the students would have understood "Trigonometric ratios of negative angles"
If you want to know more about "Trigonometric ratios of negative angles", please click here
Apart from "Trigonometric ratios of negative angles", if you need any other stuff in math, please use our google custom search here.
You can also visit our following web pages on different stuff in math.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Trigonometry word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
Recent Articles
-
Projection of Vector a On b
Oct 14, 19 08:26 AM
Projection of Vector a On b - Concept and example problems with step by step explanation
-
Solved Word Problems in Arithmetic Progression
Oct 14, 19 08:19 AM
Solved Word Problems in Arithmetic Progression - Concept and examples with step by step explanation
-
Problems on Composite Functions
Oct 14, 19 08:07 AM
Problems on Composite Functions - Concept and examples with step by step solution
