# 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 θrequired 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.

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

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

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

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

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

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

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

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6 