Product-Sum Identities

Product-sum identities, also called product-to-sum identities, rewrite products of trigonometric functions as sums or differences. They are especially useful for finding exact values, simplifying trig expressions, and proving other identities. These problems include products involving sine, cosine, tangent-related expressions, degrees, radians, and exact unit-circle values.

Notes

Notes for Product an Sum Formula

Practice Problems

\(\textbf{1)}\) Find the exact value of \(\cos\left(75^{\circ}\right)\cos\left(15^{\circ}\right)\)

 

\(\textbf{2)}\) Find the exact value of \(\sin\left(135^{\circ}\right)\cos\left(75^{\circ}\right)\)

 

\(\textbf{3)}\) Find the exact value of \(\sin\left(75^{\circ}\right)\sin\left(15^{\circ}\right)\)

 

\(\textbf{4)}\) Find the exact value of \(\cos\left(105^{\circ}\right)\cos\left(15^{\circ}\right)\)

 

\(\textbf{5)}\) Find the exact value of \(\sin\left(105^{\circ}\right)\cos\left(15^{\circ}\right)\)

 

\(\textbf{6)}\) Find the exact value of \(\cos\left(150^{\circ}\right)\cos\left(30^{\circ}\right)\)

 

\(\textbf{7)}\) Find the exact value of \(\sin\left(150^{\circ}\right)\sin\left(30^{\circ}\right)\)

 

\(\textbf{8)}\) Rewrite \(\cos(5x)\cos(2x)\) as a sum.

 

\(\textbf{9)}\) Rewrite \(\sin(6x)\sin(4x)\) as a sum or difference.

 

\(\textbf{10)}\) Rewrite \(\sin(3x)\cos(8x)\) as a sum.

 

\(\textbf{11)}\) Rewrite \(\cos(4x)\sin(x)\) as a sum.

 

\(\textbf{12)}\) Rewrite \(\cos(9x)\cos(4x)\) as a sum.

 

\(\textbf{13)}\) Rewrite \(\sin(7x)\sin(2x)\) as a sum or difference.

 

\(\textbf{14)}\) Rewrite \(\sin(10x)\cos(3x)\) as a sum.

 

\(\textbf{15)}\) Find the exact value of \(\cos\left(\frac{5\pi}{12}\right)\cos\left(\frac{\pi}{12}\right)\)

 

Challenge Problems

\(\textbf{16)}\) Find the exact value of \(\sin\left(\frac{7\pi}{12}\right)\cos\left(\frac{\pi}{12}\right)\)

 

\(\textbf{17)}\) Find the exact value of \(\sin\left(\frac{5\pi}{12}\right)\sin\left(\frac{\pi}{12}\right)\)

 

\(\textbf{18)}\) Rewrite \(2\cos(6x)\cos(4x)\) as a sum.

 

\(\textbf{19)}\) Rewrite \(2\sin(9x)\sin(5x)\) as a sum or difference.

 

\(\textbf{20)}\) Rewrite \(2\sin(8x)\cos(3x)\) as a sum.

 

See Related Pages\(\)

\(\bullet\text{ Right Triangle Trigonometry}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\)
\(\bullet\text{ Angle of Depression and Elevation}\)
\(\,\,\,\,\,\,\,\,\text{Angle of Depression}=\text{Angle of Elevation}…\)
\(\bullet\text{ Convert to Radians and to Degrees}\)
\(\,\,\,\,\,\,\,\,\text{Radians} \rightarrow \text{Degrees}, \times \displaystyle \frac{180^{\circ}}{\pi}…\)
\(\bullet\text{ Degrees, Minutes and Seconds}\)
\(\,\,\,\,\,\,\,\,48^{\circ}34’21”…\)
\(\bullet\text{ Coterminal Angles}\)
\(\,\,\,\,\,\,\,\,\pm 360^{\circ} \text { or } \pm 2\pi n…\)
\(\bullet\text{ Reference Angles}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail for Reference Angles\(…\)
\(\bullet\text{ Find All 6 Trig Functions}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail for all 6 Trig Functions\(…\)
\(\bullet\text{ Unit Circle}\)
\(\,\,\,\,\,\,\,\,\sin{(60^{\circ})}=\displaystyle\frac{\sqrt{3}}{2}…\)
\(\bullet\text{ Law of Sines}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\sin{A}}{a}=\frac{\sin{B}}{b}=\frac{\sin{C}}{c}\) Thumbnail for Law of Sines\(…\)
\(\bullet\text{ Area of SAS Triangles}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{C}\) Thumbnail for Area of SAS Triangle\(…\)
\(\bullet\text{ Law of Cosines}\)
\(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{A}\) Thumbnail for Law of Cosines\(…\)
\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\) Thumbnail for Herons Formula\(…\)
\(\bullet\text{ Geometric Mean}\)
\(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\)
\(\bullet\text{ Geometric Mean- Similar Right Triangles}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail for Geometric Mean\(…\)
\(\bullet\text{ Inverse Trigonmetric Functions}\)
\(\,\,\,\,\,\,\,\,\sin {\left(cos^{-1}\left(\frac{3}{5}\right)\right)}…\)
\(\bullet\text{ Sum and Difference of Angles Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}…\)
\(\bullet\text{ Double-Angle and Half-Angle Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(2A)}=2\sin{(A)}\cos{(A)}…\)
\(\bullet\text{ Trigonometry-Pythagorean Identities}\)
\(\,\,\,\,\,\,\,\,\sin^2{(x)}+\cos^2{(x)}=1…\)
\(\bullet\text{ Product-Sum Identities}\)
\(\,\,\,\,\,\,\,\,\cos{\alpha}\cos{\beta}=\left(\displaystyle\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\right)…\)
\(\bullet\text{ Cofunction Identities}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\cos{(\frac{\pi}{2}-x)}…\)
\(\bullet\text{ Proving Trigonometric Identities}\)
\(\,\,\,\,\,\,\,\,\sec{x}-\cos{x}=\displaystyle\frac{\tan^2{x}}{\sec{x}}…\)
\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
\(\,\,\,\,\,\,\,\,f(x)=A \sin{B(x-c)}+D \) Thumbnail for Graphing Trig Functions\(…\)
\(\bullet\text{ Solving Trigonometric Equations}\)
\(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\)

 

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