7.3 Double-Angle, Half-Angle, and Sum-to-Product Formulas
7.3.1 Learning Objectives
By the end of this section, you should be able to:
- apply double-angle formulas;
- apply power-reduction and half-angle formulas;
- determine correct half-angle signs;
- convert products to sums; and
- convert sums to products.
7.3.2 Double-Angle Formulas
\[ \sin2x=2\sin x\cos x, \]
\[ \cos2x=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x, \]
\[ \tan2x=\frac{2\tan x}{1-\tan^2x}. \]
Example 7.7 Using a Double-Angle Formula
If \(\sin x=3/5\) and \(x\) is acute, find \(\sin2x\) and \(\cos2x\).
Solution
\(\cos x=4/5\), so
\[ \sin2x=2\left(\frac35\right)\left(\frac45\right)=\frac{24}{25}, \]
\[ \cos2x=\left(\frac45\right)^2-\left(\frac35\right)^2=\frac7{25}. \]
\[ \boxed{\sin2x=24/25,\quad \cos2x=7/25} \]
Exercise 7.7 If \(\cos x=5/13\) and \(x\) is acute, find \(\cos2x\).
Show answer
Answer: \(\boxed{\cos2x=2(5/13)^2-1=-119/169}\)
7.3.3 Half-Angle and Power-Reduction Formulas
\[ \sin\frac x2=\pm\sqrt{\frac{1-\cos x}{2}}, \qquad \cos\frac x2=\pm\sqrt{\frac{1+\cos x}{2}}, \]
\[ \tan\frac x2=\frac{\sin x}{1+\cos x}=\frac{1-\cos x}{\sin x}. \]
The sign is determined by the quadrant containing \(x/2\). Power-reduction formulas are
\[ \sin^2x=\frac{1-\cos2x}{2}, \qquad \cos^2x=\frac{1+\cos2x}{2}. \]
Example 7.8 Finding an Exact Half-Angle Value
Find \(\sin22.5^\circ\).
Solution
Since \(22.5^\circ\) is in Quadrant I,
\[ \sin22.5^\circ =\sqrt{\frac{1-\cos45^\circ}{2}} =\frac{\sqrt{2-\sqrt2}}{2}. \]
\[ \boxed{\frac{\sqrt{2-\sqrt2}}{2}} \]
Exercise 7.8 Find \(\cos67.5^\circ\).
Show answer
Answer: \(\boxed{\sqrt{2-\sqrt2}/2}\)
7.3.4 Product-to-Sum and Sum-to-Product Formulas
Useful product-to-sum formulas include
\[ \sin u\cos v=\frac12[\sin(u+v)+\sin(u-v)], \]
\[ \cos u\cos v=\frac12[\cos(u+v)+\cos(u-v)], \]
\[ \sin u\sin v=\frac12[\cos(u-v)-\cos(u+v)]. \]
The corresponding sum-to-product formulas reverse these relationships.
Example 7.9 Converting a Product to a Sum
Rewrite \(\sin5x\cos2x\) as a sum.
Solution
\[ \boxed{\sin5x\cos2x=\frac12[\sin7x+\sin3x]}. \]
Exercise 7.9 Rewrite \(\cos7x+\cos3x\) as a product.
Show answer
Answer: \(\boxed{2\cos5x\cos2x}\)
7.3.5 Conceptual Takeaways
- Double-angle formulas are special cases of addition formulas.
- Equivalent cosine formulas support different algebraic goals.
- Half-angle signs depend on the half-angle’s quadrant.
- Power reduction replaces squared functions with first powers.
- Product-sum conversions change the structure of trigonometric expressions.
7.3.6 Skills You Should Be Able to Do
- Calculate double-angle values.
- Apply half-angle formulas with correct signs.
- Reduce powers of sine and cosine.
- Convert products to sums.
- Convert sums and differences to products.
7.3.7 Practice Problems with Solutions
Find \(\sin2x\) if \(\sin x=5/13\) and \(x\) is acute.
Show Solution
Since \(\cos x=12/13\),
\[ \sin2x=2(5/13)(12/13)=\boxed{120/169}. \]
Rewrite \(\cos2x\) using only \(\sin x\).
Show Solution
\[ \boxed{\cos2x=1-2\sin^2x}. \]
Find \(\cos22.5^\circ\).
Show Solution
\[ \cos22.5^\circ=\sqrt{\frac{1+\cos45^\circ}{2}} =\boxed{\frac{\sqrt{2+\sqrt2}}2}. \]
Rewrite \(\sin^2x\) using a first power of cosine.
Show Solution
\[ \boxed{\sin^2x=\frac{1-\cos2x}{2}}. \]
Rewrite \(\cos4x\cos x\) as a sum.
Show Solution
\[ \boxed{\cos4x\cos x=\frac12[\cos5x+\cos3x]}. \]
Rewrite \(\sin8x+\sin2x\) as a product.
Show Solution
\[ \sin8x+\sin2x =2\sin5x\cos3x. \]
\[ \boxed{2\sin5x\cos3x} \]