TRIGONOMETRIC RATIOS OF 90 DEGREE PLUS THETA
Trigonometric ratios of 90 degree plus theta is a part of ASTC formula in trigonometry.
Trigonometric ratios of 90 degree plus theta are given below.
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
Let us see, how the trigonometric ratios of 90 degree plus theta 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 picture shown below.
More clearly
From the above picture, it is very clear that
(90° + θ) falls in the second quadrant
In the second quadrant (90° + θ), sin and csc 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 - Examples
Example 1 :
Evaluate : sin (90° + θ)
Solution :
To evaluate sin (90° + θ), we have to consider the following important points.
(i) (90° + θ) will fall in the II nd quadrant.
(ii) When we have 90°, "sin" will become "cos".
(iii) In the II nd quadrant, the sign of "sin" is positive.
Considering the above points, we have
sin (90° + θ) = cos θ
Example 2 :
Evaluate : cos (90° + θ)
Solution :
To evaluate cos (90° + θ), we have to consider the following important points.
(i) (90° + θ) will fall in the II nd quadrant.
(ii) When we have 90°, "cos" will become "sin".
(iii) In the II nd quadrant, the sign of "cos" is negative.
Considering the above points, we have
cos (90° + θ) = - sin θ
Example 3 :
Evaluate : tan (90° + θ)
Solution :
To evaluate tan (90° + θ), we have to consider the following important points.
(i) (90° + θ) will fall in the II nd quadrant.
(ii) When we have 90°, "tan" will become "cot".
(iii) In the II nd quadrant, the sign of "tan" is negative.
Considering the above points, we have
tan (90° + θ) = - cot θ
Example 4 :
Evaluate : csc (90° + θ)
Solution :
To evaluate csc (90° + θ), we have to consider the following important points.
(i) (90° + θ) will fall in the II nd quadrant.
(ii) When we have 90°, "csc" will become "sec".
(iii) In the II nd quadrant, the sign of "csc" is positive.
Considering the above points, we have
csc (90° + θ) = sec θ
Example 5 :
Evaluate : sec (90° + θ)
Solution :
To evaluate sec (90° + θ), we have to consider the following important points.
(i) (90° + θ) will fall in the II nd quadrant.
(ii) When we have 90°, "sec" will become "csc".
(iii) In the II nd quadrant, the sign of "sec" is negative.
Considering the above points, we have
sec (90° + θ) = - csc θ
Example 6 :
Evaluate : cot (90° + θ)
Solution :
To evaluate cot (90° + θ), we have to consider the following important points.
(i) (90° + θ) will fall in the II nd quadrant.
(ii) When we have 90°, "cot" will become "tan"
(iii) In the II nd quadrant, the sign of "cot" is negative.
Considering the above points, we have
cot (90° + θ) = - tan θ
To more about ASTC formula,
After having gone through the stuff given above, we hope that the students would have understood the trigonometric ratios of 90 degree plus theta.
Apart from the stuff given in this section, 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
-
Practice Problems on Whole Numbers
Jan 18, 20 08:53 AM
Practice Problems on Whole Numbers - Practice problems in the form of test with solution
-
How to Solve Word Problems Involving Speed
Jan 18, 20 08:34 AM
How to Solve Word Problems Involving Speed - Concept and example problems with step by step solution
-
Math Olympiad Sample Problems 6th Grade
Jan 18, 20 07:33 AM
Math Olympiad Sample Problems 6th Grade - Problems with step by step solution
