Mathematics: Trigonometry(3) - High school

Main Menu

Maths High School
# h1matr3 - Trigonometry(3)

Solve the equation
Arccos x - Arcsin x	=	5π 6

	- the rule "Must have" of Mr.Zhang ®
	Identity; cos(α + β) =  cosαcosβ - sinαsinβ

Solution
From the problem, take cos both side;
Arccos x	=	Arcsin x	+	5π 6
cos(Arccos x)	=	cos(Arcsin x	+	5π 6	)
x	=	cos(Arcsin x	+	5π 6	)

From identity;

	x	=	cos (	5π 6	+	Arcsin x )
	x	=	cos	5π 6	cosθ	- sin	5π 6	sinθ

(∴  Arcsin x = θ)

-π 2 ≤ θ ≤ π 2
	θ=Arcsinx          sinθ=sin(Arcsinx)

∴ sinθ=x
cos(Arcsinx) = cos θ
                    =

	x	=	-√3 2	( √1-x2)	- ½ x

2x

=

-√3 ( √1-x2)

-  x

3x

=

-√3 ( √1-x2)

9x2

=

3 ( 1-x2)

3x2

=

( 1-x2)

4x2

=

1

x2

=

¼

x

=

(+/-)½

Verify x = ½

Arccos½=Arcsin½+

5π 6

60o=30o+150o
60o=180o	→false

Verify x = (-)½

Arccos(-)½=Arcsin(-)½+

5π 6

120o=-30o+150o
120o=120o	→right

Ans. x = - ½

Fuel Injector

Trigonometry highschool

Main Menu

Maths High School
# h1matr2 - Trigonometry

sin [ tan-1(4⁄3) + cos-1 (-12⁄13) ]

Find the value without using the tables or a calculator.

                  Strategy

- The rule "Must have" of Mr.Zhang ®

Identity; sin(α + β) =  sinαcosβ + cosαsinβ

Solution

Use the identity and inverse properties of Trigonometry;

Name	Notation	Definition	Domain of x	Range
arctangent	y=arctan x	x=tan y	all real numbers	-π/2 < y < π/2
arccosine	y=arccos x	x=cos y	-1  ≤  x ≤ 1	0  ≤  y ≤ π

tan^-1(⁴⁄₃) = arctan(⁴⁄₃) =  α
α = arctan(⁴⁄₃)
∴  tanα = ⁴⁄₃, sinα = ⁴⁄₅, cosα = ³⁄₅

cos^-1(^-12⁄₁₃) = arccos(^-12⁄₁₃) =  β
β = arccos(^-12⁄₁₃)
∴  cosβ = ^-12⁄₁₃,  sinβ = ⁵⁄₁₃

sin [ tan-1(4⁄3) + cos-1 (-12⁄13) ]	=	sin(α + β)
	=	sinα cosβ + cosα sinβ
	=	(4⁄5)(-12⁄13) + (3⁄5) (5⁄13)
Ans.	=	-33⁄65

Fuel Injector

Trigonometry

Main Menu

Maths High School
# h1matr1 - Trigonometry

A pole 5 m high is fixed on the top of a tower. The angle of elevation of

the top of the pole observed from a point 'A' on the ground is 60o and the

angle of the depression of the point "A" from the top of the tower is 45o.

Find the height of the tower.

Strategy - the rule "Must have" of Mr.Zhang ®

Solution

Compute the time to the height of tower by;

Fuel Injector