Trigonometric Identities

Contents

Figure 1

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Sine two angle addition formula

We want to prove:

\sin(x+y)=\sin x \cos y+\cos x\sin y

Looking at figure 1, note that

  • \sin(x+y)=\overline{DF}
  • \overline{DF}=\overline{DE}+\overline{EF}
  • \overline{DE}
    • \cos y = \frac{\overline{DE}}{\sin x}\quad \text{therefore,}
    • \overline{DE}=\cos y \sin x
  • \overline{EF}
    • \overline{EF}=\overline{CB}
    • \overline{CB}=\sin y
    • \sin y = \frac{\overline{CB}}{\cos x}\quad \text{therefore,}
    • \overline{CB} = \cos x \sin y \text{; because}\,\overline{CB}=\overline{EF}
    • \overline{EF}=\cos x \sin y \text{;}
  • \text{substituting values of }\overline{DF}\text{,}\,\overline{DE}\,\text{and}\,\overline{EF}\text{, we are left with}
    • \sin(x+y)=\sin x \cos y+\cos x\sin y

Cosine two angle addition formula

We want to prove

\cos(x+y)=\cos x\cos y-\sin x \sin y

Looking at figure 1, note that

  • \cos(x+y)=\overline{AF}
  • \overline{AF}=\overline{AB}-\overline{FB}
  • \overline{AB}
    • \overline{AB}=\cos y
    • \cos y=\frac{\overline{AB}}\cos x\quad \text{therefore,}
    • \overline{AB}=\cos x \cos y
  • \overline{FB}
    • \overline{FB}=\overline{EC}
    • \overline{EC}=\sin y
    • \sin y = \frac{\overline{EC}}\sin x\quad \text{therefore,}
      • \overline{EC}=\sin x \sin y
    • \overline{EC}=\overline{FB}\quad \text{thus,}
    • \overline{FB}=\sin x \sin y
  • \text{substituting values of }\overline{AF}\text{,}\,\overline{AB}\,\text{and}\,\overline{FB}\text{, we are left with}
    • \cos(x+y)=\cos x\cos y-\sin x \sin y

Sine double angle formula

We want to prove

\sin 2x=2\sin x \cos x

From the sine two angle addition formula described above,

  • \sin(x+y)=\sin x \cos y+\cos x\sin y \quad \text{then,}
  • \sin(x+x)=\sin x \cos x+\cos x\sin x=2\sin x \cos x\quad \text{of course,}
  • \sin(x+x)=\sin 2x\quad \text{therefore,}
  • \sin 2x=2\sin x \cos x

Cosine double angle formula

We want to prove

\cos 2x = \cos^2x-\sin^2x

From the cosine two angle addition formula described above,

  • \cos(x+y)=\cos x\cos y-\sin x \sin y \quad \text{then,}
  • \cos(x+x)=\cos x\cos x - \sin x \sin x = \cos^2x-\sin^2x \quad \text{of course,}
  • \cos(x+x)=\cos 2x\quad \text{therefore,}
  • \cos 2x = \cos^2x-\sin^2x

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