Simplify:
(a) [tex]$\sin \left(180^{\circ}-\theta\right)+\sin \left(360^{\circ}-\theta\right)$[/tex]
(b) [tex]$\cos \left(180^{\circ}+x\right)-\cos \left(360^{\circ}-x\right)$[/tex]
(c) [tex]$\frac{\tan \left(180^{\circ}- A \right)}{\tan \left(180^{\circ}+ A \right)}$[/tex]
(d) [tex]$\sin \left(360^{\circ}-\alpha\right) \cdot \sin \left(180^{\circ}+\alpha\right)$[/tex]
(e) [tex]$\frac{\cos \left(360^{\circ}-\theta\right) \cdot \tan \left(180^{\circ}-\theta\right)}{\cos \left(180^{\circ}-\theta\right)}$[/tex]
(f) [tex]$\frac{\sin \left(180^{\circ}- B \right) \cdot \tan \left(360^{\circ}- B \right)}{\sin \left(180^{\circ}+ B \right)}$[/tex]
(g) [tex]$\frac{\tan \left(360^{\circ}-x\right) \cdot \tan \left(180^{\circ}+x\right)}{\tan ^2\left(180^{\circ}-x\right)}$[/tex]
(h) [tex]$\frac{\sin ^2\left(360^{\circ}-\theta\right) \cdot \cos ^2\left(360^{\circ}-\theta\right)}{\sin \left(180^{\circ}+\theta\right) \cdot \sin \left(180^{\circ}-\theta\right)}$[/tex]
Answer (2)
Use trigonometric identities to simplify each expression.
Apply the identities sin ( 18 0 ∘ − x ) = sin ( x ) , sin ( 36 0 ∘ − x ) = − sin ( x ) , cos ( 18 0 ∘ + x ) = − cos ( x ) , cos ( 36 0 ∘ − x ) = cos ( x ) , tan ( 18 0 ∘ − x ) = − tan ( x ) , tan ( 18 0 ∘ + x ) = tan ( x ) , tan ( 36 0 ∘ − x ) = − tan ( x ) .
Simplify each expression by substituting the appropriate identities.
The simplified expressions are: (a) 0 , (b) − 2 cos ( x ) , (c) − 1 , (d) sin 2 ( α ) , (e) tan ( θ ) , (f) tan ( B ) , (g) − 1 , (h) − cos 2 ( θ ) .
Explanation
Stating Trigonometric Identities We will simplify the given trigonometric expressions using trigonometric identities. The key identities are:
sin ( 18 0 ∘ − x ) = sin ( x )
sin ( 36 0 ∘ − x ) = − sin ( x )
cos ( 18 0 ∘ + x ) = − cos ( x )
cos ( 36 0 ∘ − x ) = cos ( x )
tan ( 18 0 ∘ − x ) = − tan ( x )
tan ( 18 0 ∘ + x ) = tan ( x )
tan ( 36 0 ∘ − x ) = − tan ( x )
Simplifying (a) (a) sin ( 18 0 ∘ − θ ) + sin ( 36 0 ∘ − θ ) = sin ( θ ) + ( − sin ( θ )) = sin ( θ ) − sin ( θ ) = 0
Simplifying (b) (b) cos ( 18 0 ∘ + x ) − cos ( 36 0 ∘ − x ) = − cos ( x ) − cos ( x ) = − 2 cos ( x )
Simplifying (c) (c) t a n ( 18 0 ∘ + A ) t a n ( 18 0 ∘ − A ) = t a n ( A ) − t a n ( A ) = − 1
Simplifying (d) (d) sin ( 36 0 ∘ − α ) ⋅ sin ( 18 0 ∘ + α ) = ( − sin ( α )) ⋅ ( − sin ( α )) = sin 2 ( α )
Simplifying (e) (e) c o s ( 18 0 ∘ − θ ) c o s ( 36 0 ∘ − θ ) ⋅ t a n ( 18 0 ∘ − θ ) = − c o s ( θ ) c o s ( θ ) ⋅ ( − t a n ( θ )) = − c o s ( θ ) − c o s ( θ ) t a n ( θ ) = tan ( θ )
Simplifying (f) (f) s i n ( 18 0 ∘ + B ) s i n ( 18 0 ∘ − B ) ⋅ t a n ( 36 0 ∘ − B ) = − s i n ( B ) s i n ( B ) ⋅ ( − t a n ( B )) = − s i n ( B ) − s i n ( B ) t a n ( B ) = tan ( B )
Simplifying (g) (g) t a n 2 ( 18 0 ∘ − x ) t a n ( 36 0 ∘ − x ) ⋅ t a n ( 18 0 ∘ + x ) = ( − t a n ( x ) ) 2 − t a n ( x ) ⋅ t a n ( x ) = t a n 2 ( x ) − t a n 2 ( x ) = − 1
Simplifying (h) (h) s i n ( 18 0 ∘ + θ ) ⋅ s i n ( 18 0 ∘ − θ ) s i n 2 ( 36 0 ∘ − θ ) ⋅ c o s 2 ( 36 0 ∘ − θ ) = ( − s i n ( θ )) ⋅ ( s i n ( θ )) ( − s i n ( θ ) ) 2 ⋅ ( c o s ( θ ) ) 2 = − s i n 2 ( θ ) s i n 2 ( θ ) c o s 2 ( θ ) = − cos 2 ( θ )
Final Answers Therefore, the simplified expressions are: (a) 0 (b) − 2 cos ( x ) (c) − 1 (d) sin 2 ( α ) (e) tan ( θ ) (f) tan ( B ) (g) − 1 (h) − cos 2 ( θ )
Examples
Trigonometric identities are useful in various fields such as physics, engineering, and computer graphics. For example, in physics, when analyzing the motion of a pendulum or the behavior of alternating current in electrical circuits, simplifying trigonometric expressions can make the analysis easier and more intuitive. In computer graphics, trigonometric identities are used to perform rotations and transformations of objects in 3D space. Understanding these identities allows for efficient manipulation and rendering of complex scenes.
We simplified the trigonometric expressions using identities related to sine, cosine, and tangent functions. Each expression was transformed step by step into simpler forms, producing the final results of each part. The key results include zero for part (a), -2\cos(x) for part (b), and \tan(\theta) for part (e), among others.
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