Basic relationships between trigonometric functions of the same angle, Basic relationships between trigonometric functions shown in the tabular form, Basic relationships between trigonometric functions of the same angle examples

Trigonometry

Basic relationships between trigonometric functions of the same angle

Basic relationships between trigonometric functions of the same angle

Using the Pythagorean theorem for the right triangle in the

figure we get the fundamental trigonometric identity

sin² x + cos² x = 1

which, because of periodicity of the trigonometric functions,

holds for an arbitrary angle x = a + k · 2p therefore,

sin² x + cos² x = 1, x Î R.

From this identity and the definitions of the functions, tangent

and cotangent

we can derive twelve formulas so that each function is expressed through another three. Thus,

By plugging (1) and (2) into,

Dividing the basic identity by cos² x, sin² x + cos² x = 1 | ¸ cos² x

Dividing the basic identity by sin² x, sin² x + cos² x = 1 | ¸ sin² x

Basic relationships between trigonometric functions shown in the tabular form

Basic relationships between trigonometric functions of the same angle examples

Example: Find values of other trigonometric functions of an angle a, if given sin a = - 4/5 and

270° < a < 360°.

Solution: Since a is a forth-quadrant angle, then

Example: Find the value of

Solution: Dividing nominator and denominator by cos² x,

Example: Prove the identity

Solution:

Example: Given sin x + cos x = a, find sin⁴ x + cos⁴ .

Solution: Since sin² x + cos² x = 1 then,

(sin² x + cos² x)² = sin⁴ x + cos⁴ + 2sin² x · cos² x = 1 or sin⁴ x + cos⁴ = 1 - 2sin² xcos² x.

As given sin x + cos x = a then, (sin x + cos x)² = a² or sin² x + cos² x + 2sin x cos x = a²

therefore, sin x · cos x = (a² - 1)/2 and it follows that sin⁴ x + cos⁴ = 1 - (a² - 1)²/2.

Functions contents D