Trigonometrical Ratios of Complementary Angles

How to find the trigonometrical ratios of complementary angles?

If the sum of two angles is one right angle or 90°, then one angle is said to be complementary of the other. Thus, 25° and 65°; θ° and (90 - θ)° are complementary to each other.

Suppose a rotating line rotates about O in the anti-clockwise sense and starting from its initial position

Trigonometrical Ratios of Complementary Angles

\(\overrightarrow{OX}\) traces out angle ∠XOY = θ, where θ is acute. 

Take a point P on \(\overrightarrow{OY}\)  and draw \(\overline{PQ}\)  perpendicular to OX.  Let, ∠OPQ = α. Then, we have,

α + θ = 90°

or, α = 90° -  θ.

Therefore, θ and α are complementary to each other.

Now, by the definition of trigonometric ratio,

sin θ = \(\frac{\overline{PQ}}{\overline{OP}}\); ………. (i)

cos θ = \(\frac{\overline{OQ}}{\overline{OP}}\); ………. (ii)

tan θ = \(\frac{\overline{PQ}}{\overline{OQ}}\) ………. (iii)

And   sin α = \(\frac{\overline{OQ}}{\overline{OP}}\); ………. (iv)

cos α = \(\frac{\overline{PQ}}{\overline{OP}}\); ………. (v)

tan α = \(\frac{\overline{OQ}}{\overline{PQ}}\)  ….… (vi)


From (i) and (iv) we have,

sin α = cos θ   

or,  sin (90° -  θ) = cos θ;


From (ii) and (v) we have,

cos α = sin θ   

or, cos (90° -  θ) = sin θ;


From (iii) and (vi) we have,

And tan α = 1/tan θ

or, tan (90° - θ) = cot θ.


Similarly, csc (90° - θ) = sec θ;

sec (90° - θ) = csc θ

and cot (90° - θ) = tan θ.


Therefore,

Sine of any angle    = cosine of its complementary angle;

Cosine of any angle = sine of its complementary angle;

Tangent of any angle = cotangent of its complementary angle.


Corollary:

Complementary Angles: Two angles are said to be complementary if their sum is 90°. Thus θ and (90° - θ) are complementary angles.

(i) sin (90° -  θ) = cos θ

(iii) tan (90° -  θ) = cot θ

(v) sec (90° -  θ) = csc θ

(ii) cos (90° -  θ) = sin θ

(iv) cot (90° -  θ) = tan θ

(vi) csc (90° -  θ)  = sec θ

We know there are six trigonometrical ratios in trigonometry. The above explanation will help us to find the trigonometrical ratios of complementary angles.


Worked-out problems on trigonometrical ratios of complementary angles:

1. Without using trigonometric tables, evaluate \(\frac{tan  65°}{cot  25°}\)

Solution:

\(\frac{tan  65°}{cot  25°}\)

= \(\frac{tan  65°}{cot (90°  -  65°)}\)

=  \(\frac{tan   65°}{tan  65°}\), [Since cot (90° -  θ) = tan θ]

= 1


2. Without using trigonometric tables, evaluate sin 35° sin 55° - cos 35° cos 55°

Solution:

sin 35° sin 55° - cos 35° cos 55°

= sin 35° sin (90° - 35°) - cos 35° cos (90° - 35°),

= sin 35° cos 35° - cos 35° sin 35°,

                                      [Since sin (90° -  θ) = cos θ and cos (90° -  θ) = sin θ]

= sin 35° cos 35° - sin 35° cos 35°

= 0


3.  If sec 5θ = csc (θ - 36°), where 5θ is an acute angle, find the value of θ.

Solution:

    sec 5θ = csc (θ - 36°)

⇒ csc (90° - 5θ) = csc (θ - 36°), [Since sec θ = csc (90° -  θ)]

⇒ (90° - 5θ) = (θ - 36°)

⇒ -5θ - θ = -36° - 90°

⇒ -6θ = -126°

⇒ θ = 21°, [Dividing both sides by -6]

Therefore, θ = 21°


4. Using trigonometrical ratios of complementary angles prove that tan 1° tan 2° tan 3° ......... tan 89° = 1

Solution:

   tan 1° tan 2° tan 3° ...... tan 89°

= tan 1° tan 2° ...... tan 44° tan 45° tan 46° ...... tan 88° tan 89°

= (tan 1° ∙ tan 89°) (tan 2° ∙ tan 88°) ...... (tan 44° ∙ tan 46°) ∙ tan 45°

= {tan 1° ∙ tan (90° - 1°)} ∙ {tan 2° ∙ (tan 90° - 2°)} ...... {tan 44° ∙ tan (90° - 44°)} ∙ tan 45°

= (tan 1° ∙ cot 1°)(tan 2° ∙ cot 2°) ...... (tan 44° ∙ cot 44°) ∙ tan 45°, [Since tan (90° - θ) = cot θ]

= (1)(1) ...... (1) ∙ 1, [since tan θ ∙ cot θ = 1 and tan 45° = 1]

= 1

Therefore, tan 1° tan 2° tan 3° ...... tan 89° = 1

 Trigonometric Functions






11 and 12 Grade Math

From Trigonometrical Ratios of Complementary Angles 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. Successor and Predecessor | Successor of a Whole Number | Predecessor

    May 24, 24 06:42 PM

    Successor and Predecessor of a Whole Number
    The number that comes just before a number is called the predecessor. So, the predecessor of a given number is 1 less than the given number. Successor of a given number is 1 more than the given number…

    Read More

  2. Counting Natural Numbers | Definition of Natural Numbers | Counting

    May 24, 24 06:23 PM

    Natural numbers are all the numbers from 1 onwards, i.e., 1, 2, 3, 4, 5, …... and are used for counting. We know since our childhood we are using numbers 1, 2, 3, 4, 5, 6, ………..

    Read More

  3. Whole Numbers | Definition of Whole Numbers | Smallest Whole Number

    May 24, 24 06:22 PM

    The whole numbers are the counting numbers including 0. We have seen that the numbers 1, 2, 3, 4, 5, 6……. etc. are natural numbers. These natural numbers along with the number zero

    Read More

  4. Math Questions Answers | Solved Math Questions and Answers | Free Math

    May 24, 24 05:37 PM

    Math Questions Answers
    In math questions answers each questions are solved with explanation. The questions are based from different topics. Care has been taken to solve the questions in such a way that students

    Read More

  5. Estimating Sum and Difference | Reasonable Estimate | Procedure | Math

    May 24, 24 05:09 PM

    Estimating Sum or Difference
    The procedure of estimating sum and difference are in the following examples. Example 1: Estimate the sum 5290 + 17986 by estimating the numbers to their nearest (i) hundreds (ii) thousands.

    Read More