Precalculus Trigonometry

Precalculus is an advanced form of  the secondary schools algebra, is a foundational mathematical discipline. It is also called Introduction to the Analysis. In many school, precalculus is actually two separate courses: Algebra and Trigonometry.

Trigonometry is a branch of a mathematics that studies triangles, particularly right triangles. Trigonometry deal  with the relationships between the sides and the angles of triangles, and with trigonometric functions, which describe those relationships and angles in general, and the motion of waves such as sound and light waves.                                           (Source.Wikipedia)
Precalculus with Trigonometry Identities:

sin2x+ cos2 x≡ 1

1 + tan2x≡ sec2 x

1 + cot2x ≡ cosec2 x

sin2x≡ 1 – cos2x

cos2x≡ 1 – sin2x

tan2x ≡ sec2x − 1

sec2x − tan2x ≡ 1

cot2x ≡ cosec2x − 1

cosec2x − cot2x ≡ 1
precalculus trigonometry ratios:

sin (90° − x) = cosx

cos (90° − x) = sin x

tan (90° − x) = cot x

cot (90° − x) = tan x

sec (90° − x) = cosecx

cosec (90° − x) = sec x
Examples for Precalculus with Trigonometry:

Example 1:

Find x° if cosec x° = sec 55°.

Solution:

Since cosec x° =sec(90°− x°),we have sec(90°− x°)= sec 45°. ∴ 90°− x°=45°.

∴ x° = 90°− 45°= 55°.

Example 2:

Evaluate sin 30° tan 60° sec 30°

Solution:

sec 30° = sec(90°−60°) = cosec 30°=1/sin 30°

∴sin 30° tan 60° sec 70° = sin 30° tan 60° cosec 30°

=sin 30° × 3× 1/sin 30°  = 3

Examples for trigonometry identities:

Problem 1:

Prove that sin4a + cos4 a= 1 – 2sin2a cos2a.

Solution:

LHS = sin4a + cos4a = (sin2a) 2 + (cos2a) 2

= [sin2a + cos2a] 2 – 2 (sin2a) (cos2a)                 ( a2 + b2 = (a + b)2 – 2ab)

= (1)2 – 2sin2a cos2a

= 1 – 2sin2a cos2a

= RHS

Hence proved

Problem 2:

Prove that sin4b − cos4b = sin2b − cos2b

Solution:

LHS = sin4b − cos4b = (sin2b) 2 – (cos2b) 2

= (sin2b + cos2b) (sin2 b − cos2b) = (1) (sin2b − cos2b)

= sin2b − cos2b = RHS

Problem 3:

Prove that (sec c+ cos c ) (sec c − cosc ) = tan2c + sin2c.

Solution:

LHS = (sec c+ cosc ) (sec c − cosc ) = sec2c − cos2c

= (1 + tan2c) – cos2c= tan2c + (1 – cos2c)

= tan2c + sin2c = RHS