Notes

Practice Problems

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

Show Answer

The answer is \( \frac{1}{4} \)

Show Work

\(\,\,\,\,\,\cos\left(\alpha\right)\cos\left(\beta\right)=\left(\displaystyle\frac{\cos\left(\alpha + \beta\right)+\cos\left(\alpha – \beta\right)}{2}\right)\)
\(\,\,\,\,\,\cos\left(75^{\circ}\right)\cos\left(15^{\circ}\right)=\displaystyle\frac{\cos\left(75^{\circ} + 15^{\circ}\right)+\cos\left(75^{\circ} – 15^{\circ}\right)}{2}\)
\(\,\,\,\,\,\cos\left(75^{\circ}\right)\cos\left(15^{\circ}\right)=\displaystyle\frac{\cos\left(90^{\circ}\right)+\cos\left(60^{\circ}\right)}{2}\)
\(\,\,\,\,\,\cos\left(75^{\circ}\right)\cos\left(15^{\circ}\right)=\displaystyle\frac{0+ 1/2}{2}\)
\(\,\,\,\,\,\cos\left(75^{\circ}\right)\cos\left(15^{\circ}\right)=\frac{1}{4}\)

 

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

Show Answer

The answer is \( \frac{\sqrt{3}-1}{4} \)

Show Work

\(\,\,\,\,\,\sin\left(\alpha\right)\cos\left(\beta\right)=\left(\displaystyle\frac{\sin\left(\alpha + \beta\right)+\sin\left(\alpha – \beta\right)}{2}\right)\)
\(\,\,\,\,\,\sin\left(135^{\circ}\right)\cos\left(75^{\circ}\right)=\displaystyle\frac{\sin\left(135^{\circ} + 75^{\circ}\right)+\sin\left(135^{\circ} – 75^{\circ}\right)}{2}\)
\(\,\,\,\,\,\sin\left(135^{\circ}\right)\cos\left(75^{\circ}\right)=\displaystyle\frac{\sin\left(210^{\circ}\right)+\sin\left(60^{\circ}\right)}{2}\)
\(\,\,\,\,\,\sin\left(135^{\circ}\right)\cos\left(75^{\circ}\right)=\displaystyle\frac{-\frac{1}{2}+\frac{\sqrt{3}}{2}}{2}\)
\(\,\,\,\,\,\sin\left(75^{\circ}\right)\cos\left(15^{\circ}\right)=\displaystyle\frac{\sqrt{3}-1}{4}\)

 

 

See Related Pages\(\)

\(\bullet\text{ Right Triangle Trigonometry}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\)

\(\bullet\text{ Angle of Depression and Elevation}\)
\(\,\,\,\,\,\,\,\,\text{Angle of Depression}=\text{Angle of Elevation}…\)

\(\bullet\text{ Convert to Radians and to Degrees}\)
\(\,\,\,\,\,\,\,\,\text{Radians} \rightarrow \text{Degrees}, \times \displaystyle \frac{180^{\circ}}{\pi}…\)

\(\bullet\text{ Degrees, Minutes and Seconds}\)
\(\,\,\,\,\,\,\,\,48^{\circ}34’21”…\)

\(\bullet\text{ Coterminal Angles}\)
\(\,\,\,\,\,\,\,\,\pm 360^{\circ} \text { or } \pm 2\pi n…\)

\(\bullet\text{ Reference Angles}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Find All 6 Trig Functions}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Unit Circle}\)
\(\,\,\,\,\,\,\,\,\sin{(60^{\circ})}=\displaystyle\frac{\sqrt{3}}{2}…\)

\(\bullet\text{ Law of Sines}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\sin{A}}{a}=\frac{\sin{B}}{b}=\frac{\sin{C}}{c}\) \(…\)

\(\bullet\text{ Area of SAS Triangles}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{C}\) \(…\)

\(\bullet\text{ Law of Cosines}\)
\(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{A}\) \(…\)

\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\) \(…\)

\(\bullet\text{ Geometric Mean}\)
\(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\)

\(\bullet\text{ Geometric Mean- Similar Right Triangles}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Inverse Trigonmetric Functions}\)
\(\,\,\,\,\,\,\,\,\sin {\left(cos^{-1}\left(\frac{3}{5}\right)\right)}…\)

\(\bullet\text{ Sum and Difference of Angles Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}…\)

\(\bullet\text{ Double-Angle and Half-Angle Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(2A)}=2\sin{(A)}\cos{(A)}…\)

\(\bullet\text{ Trigonometry-Pythagorean Identities}\)
\(\,\,\,\,\,\,\,\,\sin^2{(x)}+\cos^2{(x)}=1…\)

\(\bullet\text{ Product-Sum Identities}\)
\(\,\,\,\,\,\,\,\,\cos{\alpha}\cos{\beta}=\left(\displaystyle\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\right)…\)

\(\bullet\text{ Cofunction Identities}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\cos{(\frac{\pi}{2}-x)}…\)

\(\bullet\text{ Proving Trigonometric Identities}\)
\(\,\,\,\,\,\,\,\,\sec{x}-\cos{x}=\displaystyle\frac{\tan^2{x}}{\sec{x}}…\)

\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
\(\,\,\,\,\,\,\,\,f(x)=A \sin{B(x-c)}+D \) \(…\)

\(\bullet\text{ Solving Trigonometric Equations}\)
\(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\)

 

In Summary

Product-sum identities or “product to sum identities” are equations that relate the product of two trigonometric functions to the sum or difference of these functions. These identities are often used to simplify expressions involving trigonometric functions, or to prove other trigonometric identities or theorems. Product-sum identities are derived from the fundamental properties of these functions, such as the double angle and half angle formulas, and the relationship between the sine and cosine.

Product-sum identities are typically introduced in a trigonometry or precalculus course, as they are an important tool for manipulating and simplifying expressions involving these functions. Product-sum identities have many applications in the real world, including calculating the distance between two points, determining the speed of an object, and computing the magnitude of a complex number in electrical engineering. Some related topics to product-sum identities include trigonometric functions, inverse trigonometric functions, and the properties of trigonometric functions. These concepts are all closely related and are important for understanding trigonometry and its applications.