4.5 Prove Trigonometric Identities

I like this sub chapter the most!
This is because the answer has given and we are sure to get correct!
Let's check it out!

In this section, we have to prove other identities by using trigonometric identities.
There are several identities which are:

Pythagorean Identities

sin^2 x + cos^2 x = 1
1+cot^2 x = cosec^2 x
1+tan^2 x = sec^2 x


Quotient Identity

tan x = sin x /cos x


Reciprocal Identities

cosec x = 1 / sin x
sec x = 1 / cos x
cot x = 1 / tan x =  cos x / sin x


Compound Angle Formulas

sin (x+y) = sin x cos y + cos x sin y
sin (x-y) = sin x cos y - cos x sin y

cos (x+y) = cos x sin y - sin x sin y

cos (x-y) = cos x sin y + sin x sin y

sin2x = 2sin x cos x
cos2x = cos^2 x -sin^2 x
           = 1 - 2sin^2 x
           = 2cos^2 x -1
tan2x = 2tan x / 1 - tan^2 x

Strategy that we need to 'survive' in proving are as below:

1. Perform substitution is based on established identities

2.Employalgebraic Manipulations:

   - factoring

   - creating common

   - denominator

             &

Lastly and it is very important

3. Common sense!!

If you are still confusing on this sub chapter

Here are videos which show how to prove of some examples

Check it out and have fun! 

* Always remember that prove the identities by starting from complicated side !! :)

4.4 Compound Angle Formulas

Overall, we have to just remember the formulas which are in this chapter.
They are additional  Formula for sine and cosine:

sin (a+b) = sin a cos b + cos a sin b
sin (a-b) = sin a cos b -cos a sin b
cos (a+b) = cos a cos b - sin a sin b
cos (a-b) = cos a cos b + sin a sin b


sin(-b) = - sin b
cos (-b) = cos b


Additional information on how the formulas will be set so:
Let's watch some videos

*Be careful with the negative sign in additional formula for cosine.

4.3 Equivalent Trigonometric Expressions

Honesty, I'm still a little bit confused of this sub chapter.

However, if we follow the formula to solve the questions, then it will be fine.

Thus, good luck ! :)

Here are some equivalent trigonometric formula:

sin x = cos (π/2 – x)	cos x = sin (π/2 – x)	sin (π/2 + x) = cosx	cos (π/2 + x) =-sinx
tan x = cot (π/2 – x)	cot x = tan (π/2 – x)	tan (π/2 + x) =-cotx	cot (π/2 + x) =-tanx
csc x = sec(π/2 – x)	sec x = csc(π/2 – x)	csc (π/2 + x ) =secx	sec (π/2 + x )=-cscx

If you are still confused to these formula, take your time and feel free to watch the video! You will learn much!

4.2 Trigonometric Ratios and Special Angles

 

Adjacent: A
Hypotenuse: H
Opposite: O

Primary Trigonometric Ratios
sin x = O / H
cos x = A / H
tan x = O / A

Reciprocal Trigonometric Ratios
cosec x = 1 / sin x
sec x = 1 / cos x
cot x = 1 / tan x

to those who are not familiar with this chapter
tan x = sin x / cos x
cot x = cos x / sin x


 Special Angles

In this chapter, we have to concern of whether our calculator is in radian mode or degree mode to avoid carelss mistakes. 

Enjoy the video below to explore more about trigonometric!

4.1 Radian Measure

 okay

now, i'm going to talk about radian measure which we should have known in secondary school if you took additional mathematics.

As a reminder:

θ = a / r 

where θ  should be in radian but not degree

a: length of arc subtended by the angle

r: radius

to convert a degree measure to a radian measure, we have to multiply the degree measure by π / 180° radians.




another stuff is angular velocity
if i am not mistaken, we didn't learn in secondary school


well,
angular velocity is the rate at which the central angle changes with respect to time


angular velocity = (degree/360° )(number of revolutions) / time in second