4.5 Prove Trigonometric Identities

The basic trigonometric identities are the Pythagorean identity, the quotient identity, the reciprocal identities, and the compound angle formulas.

To prove that an equation is an identity, treat each side of the equation independently and transform the expression on one side into exact form of the expression on the other side.

Example :

Prove that cos(π/2 - x ) = sinx

How?

L.H.S            

cos (π/2 - x ) 

= cos  π/2 cos x + sin π/2 sin x

= 0 x cos x + 1 x sin x 

R.H.S

sin x

Thus, L.H.S. = R.H.S 

Here is the video for "Prove Trigonometric Identities." Enjoy !! 

 

4.4 Compound Angle Formulas

REMEMBER !!


The compound angle, or addition and subtraction, formulas for sine and cosine are 
sin (x+y) = sinx cosy + cosx sin y
sin (x-y) = sinx cosy - cosx siny
cos (x+y) = cosx cosy - sinx siny
cos (x-y) = cosx cosy + sinx siny

Happy Friday !!

Today was the "Goth Day" for the CPU programme dress up week.
It was AMAZING !!
Most of the students wear black shirts, black pants, black dresses, black lipstick, black nail polish,...etc.
Everything is in "BLACK."
Even myself, I tried to have my first black nail polish in my life for the "Goth Day". Haha :)

"Goth Day"

It is Friday (day to have a rest), but I have a lot of homework to complete within this weekend. *headache*
It's time to stop here and continue with my work............ :)
However, we still need some entertainment to make our life better and happier. Listening music while doing homework is my favourite activity. *Enjoy*

4.3 Equivalent Trigonometric Expressions

Determine equivalent trigonometric expressions using right angle triangle :

Example :

sinx = a/b 

∠C = π/2 – x

sin ∠C = c/b

cos ∠C = a/b 

tan ∠C = c/a

Therefore,

sinx = cos ∠C , sinx = cos (π/2 – x)

Use unit circle along with transformations to derive equivalent trigonometric expressions that form other trigonometric identities, such as cos(π/2 + x ) = -sin x


Trigonometric Identities Featuring π/2

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

4.2 Trigonometric Ratios and Special Angles

*Remember*

Examples of special triangles :
(a)                                                                                     (b)

The table below is able to help you more understand about the special triangles :

♥ In this chapter, to calculate trigonometric ratios for an angle expressed in radian measure by setting the angle mode to radian in the calculator.
♥ Use the unit circle and special triangles to determine the exact values for trigonometric ratios of special angles 0,π/6, π/4, π/3, and π/2.
♥ Use the CAST rule to determine the exact values for the trigonometric ratios of multiples of the special angles.



Enjoy the VIDEOS to learn more !!

4.1 Radian Measure

♥ The radian measure of  angle θ is defined as the length, a, of the arc that subtends the angle divided by the radius,r , of the circle : θ = a/r

♥ 2π rad = 360° or π rad = 180°.

♥ To convert degree measure to radian measure, multiply the degree measure by π/180° radians.

♥ To convert radian measure to degree measure, multiply the radian measure by (180/π)°.

Example 1 : Degree Measure to Radian Measure

How ? Convert a degree measure to a radian measure, multiply the degree measure by π/180° radians.

45° = 45 x π/180°

      = π/4 radians.

Determine the approximate radian measure by using calculator.

π/4 = 3.142/4

       = 0.7855 radians.

Example 2 : Radian Measure to Degree Measure
How? Convert radian measure to degree measure, multiply the radian measure by (180/π)°
π/6 = π/6 x (180/π)°
       = 30°