Notes

Questions

Find the exact value.

\(\textbf{1)}\) \( \sin{\left(15°\right)} \)

Show Answer

The exact value is \( \displaystyle \frac{\sqrt{6}-\sqrt{2}}{4} \)

Show Work

\(\text{Step 1: Rewrite } \sin\left(15°\right) \text{ in terms of angles measures we know.} \)

\(\,\,\,\,\,\,\sin\left(15°\right)=\sin\left(60°-45°\right)\)

\(\text{Step 2: Use notes to expand.} \)

\(\,\,\,\,\,\,\sin\left(A-B\right)=\sin\left(A\right)\cos\left(B\right)-\cos\left(A\right)\sin\left(B\right)\)

\(\,\,\,\,\,\,\sin\left(60°-45°\right)=\sin\left(60°\right)\cos\left(45°\right)-\cos\left(60°\right)\sin\left(45°\right)\)

\(\text{Step 3: Evaluate.} \)

\(\,\,\,\,\,\,\sin\left(60°-45°\right)=\displaystyle\left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)-\left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right)\)

\(\text{Step 4: Simplify.} \)

\(\,\,\,\,\,\,\sin\left(60°-45°\right)=\displaystyle\left(\frac{\sqrt{6}}{4}\right)-\left(\frac{\sqrt{2}}{4}\right)\)

\(\,\,\,\,\,\,\sin\left(60°-45°\right)=\displaystyle\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)\)

\(\textbf{2)}\) \( \cos{\left(195°\right)} \)

Show Answer

The exact value is \( \displaystyle -\frac{\sqrt{6}+\sqrt{2}}{4} \)

Show Work

\(\text{Step 1: Rewrite } \cos\left(195°\right) \text{ in terms of angles measures we know.} \)

\(\,\,\,\,\,\,\cos\left(195°\right)=\cos\left(150°+45°\right)\)

\(\text{Step 2: Use notes to expand.} \)

\(\,\,\,\,\,\,\cos\left(A+B\right)=\cos\left(A\right)\cos\left(B\right)-\sin\left(A\right)\sin\left(B\right)\)

\(\,\,\,\,\,\,\cos\left(150°+45°\right)=\cos\left(150°\right)\cos\left(45°\right)-\sin\left(150°\right)\sin\left(45°\right)\)

\(\text{Step 3: Evaluate.} \)

\(\,\,\,\,\,\,\cos\left(150°+45°\right)=\displaystyle\left(-\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)-\left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right)\)

\(\text{Step 4: Simplify.} \)

\(\,\,\,\,\,\,\cos\left(150°+45°\right)=\displaystyle\left(\frac{-\sqrt{6}}{4}\right)-\left(\frac{\sqrt{2}}{4}\right)\)

\(\,\,\,\,\,\,\cos\left(195°\right)=\displaystyle -\frac{\sqrt{6}+\sqrt{2}}{4}\)

\(\textbf{3)}\) \( \cos⁡{\left(75°\right)} \)

Show Answer

The exact value is \( \displaystyle \frac{\sqrt{6}-\sqrt{2}}{4} \)

Show Work

\(\text{Step 1: Rewrite } \cos{\left(75°\right)} \text{ in terms of angles measures we know.} \)

\(\,\,\,\,\,\,\cos{\left(75°\right)} = \cos{\left(30°+45°\right)}\)

\(\text{Step 2: Use notes to expand.} \)

\(\,\,\,\,\,\,\cos{\left(A+B\right)} = \cos{\left(A\right)}\cos{\left(B\right)} – \sin{\left(A\right)}\sin{\left(B\right)}\)

\(\,\,\,\,\,\,\cos{\left(30°+45°\right)} = \cos{\left(30°\right)}\cos{\left(45°\right)} – \sin{\left(30°\right)}\sin{\left(45°\right)}\)

\(\text{Step 3: Evaluate.} \)

\(\,\,\,\,\,\,\cos{\left(30°+45°\right)} = \displaystyle\left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) – \displaystyle\left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right)\)

\(\text{Step 4: Simplify.} \)

\(\,\,\,\,\,\,\cos{\left(30°+45°\right)} = \displaystyle\left(\frac{\sqrt{6}}{4}\right) – \displaystyle\left(\frac{\sqrt{2}}{4}\right)\)

\(\,\,\,\,\,\,\cos{\left(30°+45°\right)} = \displaystyle\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)\)

\(\textbf{4)}\) \( \sin{\left(285°\right)} \)

Show Answer

The exact value is \( \displaystyle -\frac{\sqrt{6}+\sqrt{2}}{4} \)

\(\textbf{5)}\) \( \cos{\left(-15°\right)} \)

Show Answer

The exact value is \( \displaystyle \frac{\sqrt{6}+\sqrt{2}}{4} \)

\(\textbf{6)}\) Simplify \( \displaystyle \frac{\tan{\frac{11\pi}{8}-\tan{\frac{3\pi}{8}}}}{1+\tan{\frac{11\pi}{8}-\tan{\frac{3\pi}{8}}}} \)

Show Answer

The answer is \( 0 \)

Verify the following identity.

\(\textbf{7)}\) \( \cos⁡{\left(90°+θ\right)}=-\sin⁡{θ} \)

Show Steps

\(\,\,\,\,\, \cos⁡{\left(90°+θ\right)}=-\sin⁡{θ} \,\,\,\,\, \left(\text{Given}\right) \)

\(\,\,\,\,\, \cos⁡{90}\cos{θ}-\sin{90}\sin{θ}=-\sin⁡{θ} \,\,\,\,\, \left(\text{Sum Angle Formula}\right) \)

\(\,\,\,\,\, (0)\cos{θ}-(1)\sin{θ}=-\sin⁡{θ} \,\,\,\,\, \left(\sin{90}=1 \text{ and } \cos(90)=0\right) \)

\(\,\,\,\,\, \sin{θ}=-\sin⁡{θ} \,\,\,\,\, \left(\text{simplify}\right) \)

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

Sum and difference of angles formulas are a great way to find the exact value of sine or cosine for a lot of angles that don’t show up on the unit circle, but can be found by adding or subtracting two angles from the unit circle. For example \(75^{\circ}=120^{\circ}-45^{\circ}\).