Trigonometry Calculating Range

Introduction :
In mathematics trigonometry is one of the divisions. Trigonometry is derivative from the word Greek. In trigonometry, mainly study about the right triangle. Angle and side of the triangle is deal is deals with trigonometry.  The spherical trigonometry is major division of the trigonometry.  Interval of the given value (higher value and lower values) is said to be range. In the following we see detailed about trigonometry calculating range.

Range of Trigonometry Function:

The different between input values is said to be range. Range between the smaller value and the higher value.

Sine function:

[-1, 1] is the range of the sine function

Cosine function:

[-1, 1] is range of the cosine function.

Tangent function:

All real number is the range of the tangent function.

Cotangent:

All the real number is range of the cotangent.

Secant function:

Range of secant function is the [-`oo` ,-1] and [1,+`oo` ] is ;

Cosecant function:

The range of cosecant function is [-`oo` , -1] and [1, + `oo` ]

Example1: Trigonometry Calculating Range

Find the range of the given function:

P=13sin`(x+(Pi)/(8))+5`

Solution:

In this problem given function is:

P=13sin`(x+(Pi)/(8))+5`

For

P=sin`theta`

Common range value of the given sin function is:

-1`<=sintheta<=+1`

`theta``(x+(pi)/(8)) `

Substitute the `theta` value.

-1`<=sin(x+(pi)/(8))<=+1`

Multiply 13 from the both sides:

-13`<=13sin(x+(pi)/(8))<=+13`

Adding +5 from both side:

-13+5`<=13sin(x+(pi)/(8)+5)<=+13+5`

-8`<=13sin(x+(pi)/(8)+5)<=+18`

Range of the given problem is [-8, 18]

Example2: Trigonometry Calculating Range

Find the range of the given function:

P = 9sin`(x+(Pi)/(7))+3`

Solution:

In this problem given function is:

P = 9sin`(x+(Pi)/(7))+3`

For P = sin`theta`

Common range value of the given sin function is:

-1`<=sintheta<=+1`

`theta``(x+(pi)/(7)) `

Substitute the `theta` value.

-1`<=sin(x+(pi)/(7))<=+1`

Multiply 9 from the both sides:

-9`<=9sin(x+(pi)/(7))<=+9`

Adding +3 from both side:

-9+3`<=9sin(x+(pi)/(7))+3<=+9+3`

-6`<=9sin(x+(pi)/(7)+3)<=+12`

Range of the given problem is [-6, 12]

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Example3: Trigonometry Calculating Range

Find the range of the given function:

P = 6sin`(x+(Pi)/(4))+2`

Solution:

In this problem given function is:

P = 6sin`(x+(Pi)/(4))+2`

For P = sin`theta`

Common range value of the given sin function is:

-1`<=sintheta<=+1`

`theta``(x+(pi)/(4)) `

Substitute the `theta` value.

-1`<=sin(x+(pi)/(4))<=+1`

Multiply 6 from the both sides:

-6`<=6sin(x+(pi)/(4))<=+6`

Adding +2 from both side:

-6+2`<=6sin(x+(pi)/(4))+2<=+6+2`

-4`<=6sin(x+(pi)/(4))+2<=+8`

Range of the given problem is [-4, 8]

Trig Word Problems

Introduction to trig word problems:

Greek Mathematician Ptolemy, Father of trigonometry proved the equation sin2A+cos2A=1 using geometry involving a relationship between the chords of a circle. Trigonometry was mainly concerned with establishing the relations between sides and angles of a triangle. The trigonometry consists of angles, quadrants, ratios and Identities, Compound Angles, and trigonometrical Equations. The trigonometry word example problems and practice problems are given below.

Example Problems for Trig Word Problems:

Trig word problems - Example: 1

The top of a tower was seen from the top and the bottom of a building of height 10 m at angles of elevation 45° and 60°. Determine the height of the tower.

Solution:

 

Let AB be the tower CD be the building of height 10 m.

Let DE be perpendicular to AB from D.

At C, the angle of elevation of A is 60°.

That is ?BCA = 60°

At D, the angle of elevation of A is 45°.

That is ?EDA = 45°

BE = CD = 10 m (since BEDC is a rectangle)

Let AE be x in metres, then AB is = x +10 m

In a right angled triangle ABC

tan60° = AB/BC (or) v3 = (x + 10m)/BC

After solve this, we get

ED = BC = (x + 10m)/v3

In a right angled triangle AED

tan45° = AE/ED   (or)

v3 x = x + 10 m

v3 x – x = 10 m

x( v3 –1) = 10 m

After solving this, we get

= 5 (1.732+1)m = 5 (2.732) m = 13.66 m

Height of the tower AB = x + 10 m = 13.66 m + 10 m = 23.66 m

Trig word problems - Example: 2

Determine the length of the chord of a circle of radius 10 cm subtending at the centre the angle of 144°.

Solution:

Let AB be a chord of a circle with centre 0 of radius 10 cm. Draw OC?AB. Then C is the mid point of AB

?AOB = 144°

?COB = 72°

In right angled triangle OCB

OB/BC = sin 72°

BC = 10 sin 72° cm

= 10 × 0.9511 cm

= 9.511 cm

Length of chord AB = 2 × BC

= 2 × 9.511 cm =  19.022cm . Is this topic Translation Math hard for you? Watch out for my coming posts.

Practice for Trigonometry Word Problems:

1. A flag staff stands on the top of 6 m high tower. From a point on the ground the angle of elevation of given top of the flag staff is 60° and from the same point the angle of elevation of the top of the tower is 45°. Determine the height of the flag staff.

Answer: 4.392

2. A ladder placed that against the wall such that, the ladder are reaches the top of the wall then height 6 m and the ladder is inclined at an angle of 60°. Determine how far the ladder is from the foot of the wall.

Answer: 3.464