Trigonometry Gr 11 Identities Revision

REVISION: GRADE 11 TRIG IDENTITIES AND REDUCTION FORMULAE

Identities

sin A
tan A = cos A

sin2 A + cos2 A = 1
Hence
sin2 A = 1 − cos2 A
and
cos2 A = 1 − sin2 A

180° rule and negative angles

sin( 180° − A) = sin A cos( 180° − A) = − cos A tan( 180° − A) = − tan A
sin( 180° + A) = − sin A cos( 180° + A) = − cos A tan( 180° + A) = tan A
sin( 360° − A) = − sin A cos( 360° − A) = cos A tan( 360° − A) = − tan A
sin( − A) = − sin A cos( − A) = cos A tan( − A) = − tan A

90° rule

sin( 90° − A) = cos A
cos( 90° − A) = sin A

Special angles

We use our knowledge of the standard graphs for sin, cos and tan of values like 0°, 90°,
180° and 360°, and the special triangles for reductions that lead to 30°, 45° and 60°.

(90;1) (0;1) (360;1)
1
(45;1). 45° √2 2
(360;0) 30°
(360;0) (270;0) (0;0) 1 √3
(180;0) (90;0) (180;0)
45° 60°
-1 1 1
(270;-1) (180;-1)




1 1 √3
sin 0° = 0 sin 30° = sin 45° = sin 60° = sin 90° = 1 sin 180° = 0 sin 270 ° = −1 sin 360° = 0
2 √2 2

√3 1 1
cos 0° = 1 cos 30° = cos 45° = cos 60° = cos 90° = 0 cos 180° = −1 cos 270 ° = 0 cos 360° = 1
2 √2 2

1 √3 tan 90° is tan 270° is
tan 0° = 0 tan 30° = tan 45° = 1 tan 60° = tan 180° = 0 tan 360° = 0
√3 2 undefined undefined




1