Today, in this article I'm gonna show to you how to solve a Trigonometric Functions of the Two Angles. This might help you in your math subject especially for those who hates Mathematics.
1.) Find the exact value of of sin75°
Solution:
sin(75°) = sin(30°+45°)
Using this formula; sin(A+B) = sinAcosB + cosAsinB
Given:
sinA = sin30°= ½
cosB = cos45°= √2/2
cosA = cos30°= √3/2
sinB = sin45°= √2/2
sin(30°+45°) = sin30°cos45° + cos30°sin45°
sin(30°+45°) = (1/2)(√2/2) + (√3/2)(√2/2)
sin(30°+45°) = (√2/4) + (√6/4)
sin(75°) = (√2 + √6)/4
2. Find the exact value of of tan15°
Solution:
tan15° = tan(60° -45°)
Using this formula tan(A-B) = (tanA -tanB)/(1 +tanAtanB)
Given:
tanA = tan60° = √3
tanB = tan45° = 1
tan(60° -45°) = (√3 -1)/(1 +√3)
tan(60° -45°) = [(√3 -1)/(1 +√3)]*[(1-√3)/(1-√3)] -----> (Rationalize the denominator)
tan(60° -45°) = (2√3 -4)/(-2)
tan15° = -2√3 +2
3. If sinA=2/3, find sin(A-90°).
Solution:
Using Subtraction Formula; sin(A-B) = sinAcosB - cosAsinB
Since sinA=2/3 = o/h, we’re going to solve for “a” using Phythagorean theorem
h^2 = a^2 +o^2
3^2 = a^2 +2^2
a = √5
sin(A-90°) = sinAcos90° - cosAsin90°
Given:
sinA =2/3
cosA =a/h =√5/3 --------> (substitute the value of “a” we have solve ealier)
cos90° =0
sin90° =1
.
sin(A-90°) = (2/3)*0 – (√5/3)
sin(A-90°) = -√5/3
4. Given sinA =4/5 and cosB =5/13, find sin(A+B) +sin(A-B).
Solution:
Simplify first sin(A+B) +sin(A-B)
sin(A+B) +sin(A-B) = (sinAcosB +cosAsinB)+(sinAcosB -cosAsinB)
sin(A+B) +sin(A-B) = 2sinAcosB
2sinAcosB
2(4/5)(5/13)
(substitute the given values)
= 8/13
5. Find the exact value of (tan73° +tan32°)/(1 –tan73°tan32°).
Solution:
Notice the given expression above is in the form of “(tanA +tanB)/(1 -tanAtanB)” which is equal to tan(A+B) thus,
tan(73°+32°) = (tan73° +tan32°)/(1 –tan73°tan32°).
tan(73°+32°) = tan(105°) =tan(60°+45°) --------> (we rewrite the given angle as the sum of two special angle)
tan(60°+45°) = (tan60° +tan45°)/(1 –tan60°tan45°)
Given:
tan(45°) =1
tan(60°) =√3
tan(60°+45°) = (tan60° +tan45°)/(1 –tan60°tan45°)
tan(60°+45°) = (√3 +1)/(1 –√3)
tan(60°+45°) = [(√3 +1)/(1 –√3)]*[( 1 +√3)/( 1 +√3)] --------> (Rationalize the Denominator)
tan(60°+45°) = (2√3 +4)/(-2)
.
= -√3 -2
Conclusion
I know Maths really is hard to understand but if you pursue and enjoy it you will get used to it and you can learn more easily.