Tha value of sin 78°+cos132°is
Answers
Answer:
(√5-1)/4
Step-by-step explanation:
sin 78°+cos 132° = sin 78°+ cos(90+42)°
= sin 78° - sin 42°
= 2cos(120°/2)sin(36°/2) [sin C-sinD =2cos(C+D)/2 sin(C-D)/2]
= 2cos 60° sin 18°
= 2×(1/2)×(√5-1)/4
= (√5-1)/4
Concept-
Trigonometric identities are the equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality can be defined.
Given-
Expression is given as sin 78° + cos 132°
Find-
Find the value of sin 78° + cos 132°
Solution-
sin 78° + cos 132°
Use the identity : cos(x) = sin ( 90° - x )
cos (132°) = sin (90° - 132°)
sin (78°) + sin (90° - 132°)
Simplify-
sin (78°) + sin ( -42°)
Use the following property : sin ( -x ) = - sin ( x )
sin (-42°) = - sin (42°)
sin (78°) - sin (42°)
Use the following identity : sin (s) - sin (t) = 2 cos (s + t)/2 sin (s - t)/2
sin (78°) - sin (42°) = 2 cos (78° + 42°)/2 . sin (78° - 42°)/2
2 cos (78° + 42°)/2 sin (78 °- 42°)/2
Simplify -
2 cos (60°) sin (18°)
Use the following identity -
cos (60°) = 1/2
sin (18°) = [-2 + √2² - 4 . 4 (-1) ] / 2.4
sin 18° = (-2 + √20) / 2.4
sin 18° = (-2 + 2√5) /2.4
sin 18° = (√5 - 1) / 4
⇒ 2 cos (60°) sin (18°)
⇒ 2 ( 1/2) . (√5 - 1) /4
⇒ 1 . (√5 - 1)/4
⇒ (√5 - 1 ) / 4
Therefore , the value of sin 78° + cos132° is (√5 - 1 ) / 4.
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