Proofs of Fundamental Trigonometric Identities hy
pote 𝑐 𝑏 nuse
leg 𝜃 𝑎 leg opposite
𝑏
sin 𝜃 hypotenuse
𝑐 = 𝑏 ÷ 𝑎 = 𝑏 ∙ 𝑐 = 𝑏 = opposite = tan 𝜃 tan 𝜃 =
=
=𝑎
adjacent
cos 𝜃
𝑐 𝑐 𝑐 𝑎 𝑐 adjacent
𝑐
hypotenuse
adjacent
𝑎
cos 𝜃 hypotenuse
𝑎 𝑏 𝑎 𝑐 𝑎 adjacent
cot 𝜃 =
=
= 𝑐 = ÷ = ∙ = =
= cot 𝜃 opposite
𝑏 𝑐 𝑐 𝑐 𝑏 𝑏 opposite
sin 𝜃
hypotenuse
𝑐
!
Proof of sin 𝜃 + cos ! 𝜃 = 1 Pythagorean Theorem 𝑎! + 𝑏 ! = 𝑐 ! Divide by 𝑐 ! 𝑎! 𝑏! 𝑐 !
+
= 𝑐! 𝑐! 𝑐!
Reduce 𝑎
𝑐
!
Apply Trigonometric Definitions adjacent
hypotenuse
!
+
𝑏
𝑐
!
= 1 opposite
+
hypotenuse
Apply Trigonometric Definitions cos ! 𝜃 + sin! 𝜃 = 1 Additive Identity sin! 𝜃 + cos ! 𝜃 = 1 !
= 1