Sum, Difference & Product Formulae

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SUMMARY

The forum discussion centers on the trigonometric identity \(\sin10^{\circ}+\sin80^{\circ}=\sqrt{2}\cos35^{\circ}\). The user attempts to derive this identity using the sum-to-product formulas, specifically \(2\sin(\frac{90}{2})\cos(\frac{-70}{2})\). The solution process reveals a sign error, leading to the incorrect conclusion of \(-\sqrt{2}\cos35^{\circ}\) instead of the correct positive form. The discussion emphasizes the importance of correctly applying trigonometric identities and recognizing the properties of cosine.

PREREQUISITES
  • Understanding of trigonometric identities, specifically sum-to-product formulas.
  • Familiarity with the unit circle and angle measures in degrees.
  • Basic algebraic manipulation skills for simplifying trigonometric expressions.
  • Knowledge of cosine properties, particularly \(\cos(-x) = \cos(x)\).
NEXT STEPS
  • Study the derivation of sum-to-product identities in trigonometry.
  • Practice solving trigonometric equations without a calculator.
  • Explore the implications of angle properties in trigonometric functions.
  • Review common mistakes in trigonometric simplifications and how to avoid them.
USEFUL FOR

Students studying trigonometry, educators teaching trigonometric identities, and anyone looking to improve their problem-solving skills in mathematics.

odolwa99
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My answer is almost correct, except for the negative sign. Can anyone help?

Many thanks.

Homework Statement



Q. Without using a calculator, show that \sin10^{\circ}+\sin80^{\circ}=\sqrt{2}\cos35^o

Homework Equations



The Attempt at a Solution



2\sin(\frac{90}{2})\cos(\frac{-70}{2})
-2\sin45\cos35
-2(\frac{1}{\sqrt{2}})\cos35
\frac{-2}{\sqrt{2}}\cos35
\frac{-2\sqrt{2}}{2}\cos35
-\sqrt{2}\cos35
 
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odolwa99 said:
My answer is almost correct, except for the negative sign. Can anyone help?

Many thanks.

Homework Statement



Q. Without using a calculator, show that \sin10^{\circ}+\sin80^{\circ}=\sqrt{2}\cos35^o

Homework Equations



The Attempt at a Solution



2\sin(\frac{90}{2})\cos(\frac{-70}{2})
-2\sin45\cos35

##\cos(-35)=+\cos(35)##
 
Thank you.
 

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