Trigonometrical Ratios of (90° + θ)

What is the relation among all the trigonometrical ratios of (90° + θ)?

In trigonometrical ratios of angles (90° + θ) we will find the relation between all six trigonometrical ratios.

Let a rotating line OA rotates about O in the anti-clockwise direction, from initial position to ending position makes an angle ∠XOA = θ again the same rotating line rotates in the same direction and makes an angle ∠AOB =90°.

Trigonometrical Ratios of (90° + θ)

Diagram 1

Trigonometrical Ratios of (90° + θ)

Diagram 2

Trigonometrical Ratios of (90° + θ)

Diagram 3

Trigonometrical Ratios of (90° + θ)

Diagram 4

Therefore we see that, ∠XOB = 90° + θ

Take a point C on OA and draw CD perpendicular to OX or OX’.

Again, take a point E on OB such that OE = OC and draw EF perpendicular to OX or OX’. From the right-angled ∆ OCD and ∆ OEF we get,

∠COD = ∠OEF [since OB ⊥ OA]

and OC = OE.

Therefore, ∆ OCD ≅ ∆ OEF (congruent).

Therefore according to the definition of trigonometric sign, OF = - DC, FE = OD and OE = OC

We observe that in diagram 1 and 4 OF and DC are opposite signs and FE, OD are either both positive. Again we observe that in diagram 2 and 3 OF and DC are opposite signs and FE, OD are both negative.

According to the definition of trigonometric ratio we get,

sin (90° + θ) = FEOE

sin (90° + θ) = ODOC, [FE = OD and OE = OC, since ∆ OCD ≅ ∆ OEF]

sin (90° + θ) = cos θ


cos (90° + θ) = OFOE

cos (90° + θ) = DCOC, [OF = -DC and OE = OC, since ∆ OCD ≅ ∆ OEF]

cos (90° + θ) = - sin θ.


tan (90° + θ) = FEOF

tan (90° + θ) = ODDC, [FE = OD and OF = - DC, since ∆ OCD ≅ ∆ OEF]

tan (90° + θ) = - cot θ.


Similarly, csc (90° + θ) = 1sin(90°+Θ)

csc (90° + θ) =  1cosΘ

csc (90° + θ) = sec θ.


sec (90° + θ) = 1cos(90°+Θ)

sec (90° + θ) =  1sinΘ

sec (90° + θ) = - csc θ.


and cot (90° + θ) = 1tan(90°+Θ)

cot (90° + θ) = 1cotΘ

cot (90° + θ) = - tan θ.


Solved examples:

1. Find the value of sin 135°.

Solution:

sin 135° = sin (90 + 45)°

            = cos 45°; since we know, sin (90° + θ) = cos θ

            = 12


2. Find the value of tan 150°.

Solution:

tan 150° = tan (90 + 60)°

            = - cot 60°; since we know, tan (90° + θ) = - cot θ

            = 13

 Trigonometric Functions





11 and 12 Grade Math

From Trigonometrical Ratios of (90° + θ) to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Volume of a Cube | How to Calculate the Volume of a Cube? | Examples

    Jul 22, 25 03:02 PM

    Volume of a Cube
    A cube is a solid box whose every surface is a square of same area. Take an empty box with open top in the shape of a cube whose each edge is 2 cm. Now fit cubes of edges 1 cm in it. From the figure i…

    Read More

  2. Volume of a Cuboid | Volume of Cuboid Formula | How to Find the Volume

    Jul 20, 25 12:58 PM

    Volume of Cuboid
    Cuboid is a solid box whose every surface is a rectangle of same area or different areas. A cuboid will have a length, breadth and height. Hence we can conclude that volume is 3 dimensional. To measur…

    Read More

  3. 5th Grade Volume | Units of Volume | Measurement of Volume|Cubic Units

    Jul 20, 25 10:22 AM

    Cubes in Cuboid
    Volume is the amount of space enclosed by an object or shape, how much 3-dimensional space (length, height, and width) it occupies. A flat shape like triangle, square and rectangle occupies surface on…

    Read More

  4. Worksheet on Area of a Square and Rectangle | Area of Squares & Rectan

    Jul 19, 25 05:00 AM

    Area and Perimeter of Square and Rectangle
    We will practice the questions given in the worksheet on area of a square and rectangle. We know the amount of surface that a plane figure covers is called its area. 1. Find the area of the square len…

    Read More

  5. Area of Rectangle Square and Triangle | Formulas| Area of Plane Shapes

    Jul 18, 25 10:38 AM

    Area of a Square of Side 1 cm
    Area of a closed plane figure is the amount of surface enclosed within its boundary. Look at the given figures. The shaded region of each figure denotes its area. The standard unit, generally used for…

    Read More