PROVING PROBLEMS INVOLVING TRIGONOMETRIC ANGLES FOR GRADE 11

Problem 1 :

Prove that

[cot(180° + θ)sin(90° - θ)cos(- θ)]/[sin(270° + θ)cot(180° + θ)tan (- θ)cosec(360° + θ)  =  cos2θcotθ 

Solution :

cot(180° + θ) : 

Lies in 3rd quadrant. For tan and cot we will have positive sign. 

cot(180° + θ)   =  -cotθ 

sin(90° - θ) :

Lies in 1st quadrant. For all trigonometric ratios, we will have positive sign. 

sin(90° - θ)  =  cosθ 

cos(- θ) : 

According to the property cos (- θ)  =  cos θ

sin(270° + θ) :

Lies in 4th quadrant. For cos and sec we will have positive sign. 

sin(270° + θ)  =  cosθ 

tan(- θ) :

According to the property tan(- θ)  =  -tanθ

cosec(360° + θ) :

Lies in 1st quadrant. For all trigonometric ratios, we will have positive sign. 

cosec(360° + θ)  =  cosecθ 

Then, 

[cot(180° + θ)sin(90° - θ)cos(- θ)]/[sin(270° + θ)cot(180° + θ) tan (- θ)cosec(360° + θ) :

=   -cot θ cos θ cos θ /cos θ (-tan θ) cosec θ 

=  (cot θ cos θ) / (tan θ cosec θ)

=   cosθ cot θ

Problem 2 :

Find all the angles between 0° and 360°  which satisfy the equation sin2θ = 3/4.

Solution :

sin2θ  =  3/4

sinθ  =  √(3/4)

sinθ  =  3/2

 θ  =  sin-1 3/2

 θ  =  π/3

In 2nd quadrant, we will have positive values for the trigonometric ratios sin θ and cosec θ

required angle  =  π - (π/3)

  =  2π/3

Hence the required angles are π/3 and 2π/3.

Problem 3 :

Show that sin2 π/18 + sin2 π/9 + sin2 7π/18 + sin2 4π/9  =  2.

Solution :

 sin2π/18 + sin2π/9 + sin27π/18 + sin24π/9 :

=  (sin π/18)2 + (sin π/9)2 + (sin 7π/18)2 + (sin 4π/9)2

=  sin210 + sin220 + sin 270 + sin280

=  [cos(90 - 10)]2 + [cos(90 - 20)] + sin 2 70 + sin2 80 

=  cos280 + cos270  + sin270 + sin280

 =  sin280 + cos280 + sin270 + cos270

 =  1 + 1

=  2

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. SAT Math Videos

    May 22, 24 06:32 AM

    sattriangle1.png
    SAT Math Videos (Part 1 - No Calculator)

    Read More

  2. Simplifying Algebraic Expressions with Fractional Coefficients

    May 17, 24 08:12 AM

    Simplifying Algebraic Expressions with Fractional Coefficients

    Read More

  3. The Mean Value Theorem Worksheet

    May 14, 24 08:53 AM

    tutoring.png
    The Mean Value Theorem Worksheet

    Read More