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Notes

Intro

An oblique triangle is any triangle that is not a right triangle. To find the area of these triangles you would use either Heron’s Formula or the Law of sines formula as shown above.

Questions

\(\textbf{1)}\) Find the area of this triangle.

Show Area

The area is \(= \displaystyle\frac{15\sqrt{3}}{4} \approx 6.4952 \) units squared

Show Work

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{(C)}\)

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}(5)(3) \sin{(120)}\)

\(\,\,\,\,\,\,\text{Area}=\frac{15}{2} \cdot \frac{\sqrt{3}}{2}\)

\(\,\,\,\,\,\,\text{Area}= \displaystyle\frac{15\sqrt{3}}{4} \approx 6.4952\) units squared

\(\textbf{2)}\) Find the area of this triangle.

Show Answer

\(\text{The area is }\approx 19.90\)

Show Work

\(\text{Step 1: Find Semiperimeter}\)

\(\,\,\,\,\,\,s=\displaystyle \frac{a+b+c}{2}\)

\(\,\,\,\,\,\,s=\displaystyle \frac{5+8+9}{2}\)

\(\,\,\,\,\,\,s=\displaystyle 11\)

\(\text{Step 2: Find the Area}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{s(s-a)(s-b)(s-c)}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{11(11-9)(11-8)(11-5)}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{11(2)(3)(6)}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{396}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle 6\sqrt{11}\)

\(\text{The area is }\approx 19.90\) units squared

\(\textbf{3)}\) Find the area of this triangle.

Show Area

The area is \(\approx 10.5497 \) units squared

Show Work

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{(C)}\)

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}(6)(9) \sin{(23)}\)

\(\,\,\,\,\,\,\text{Area}=27 \cdot \left(0.39073\right)\)

\(\,\,\,\,\,\,\text{Area}\approx 10.5497\) units squared

\(\textbf{4)}\) Find the area of this triangle.

Show Answer

The area is \(\approx13.42\) units squared

Show Work

\(\text{Step 1: Find Semiperimeter}\)

\(\,\,\,\,\,\,s=\displaystyle \frac{a+b+c}{2}\)

\(\,\,\,\,\,\,s=\displaystyle \frac{9+7+4}{2}\)

\(\,\,\,\,\,\,s=\displaystyle 10\)

\(\text{Step 2: Find the Area}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{s(s-a)(s-b)(s-c)}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{10(10-9)(10-7)(10-4)}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{10(1)(3)(6)}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{180}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle 3\sqrt{20}\)

\(\text{The area is }\approx13.42\) units squared

\(\textbf{5)}\) Find the area of this triangle.

Show Area

The area is \(\approx 11.2763 \) units squared

Show Work

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{(C)}\)

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}(3)(8) \sin{(70)}\)

\(\,\,\,\,\,\,\text{Area}=12 \cdot \left(0.93969\right)\)

\(\,\,\,\,\,\,\text{Area}\approx 11.2763\) units squared

\(\textbf{6)}\) Find the area of this triangle.

Show Answer

\(\text{The area is }\approx 35.14\)

Show Work

\(\text{Step 1: Find Semiperimeter}\)

\(\,\,\,\,\,\,s=\displaystyle \frac{a+b+c}{2}\)

\(\,\,\,\,\,\,s=\displaystyle \frac{6+14+18}{2}\)

\(\,\,\,\,\,\,s=\displaystyle 19\)

\(\text{Step 2: Find the Area}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{s(s-a)(s-b)(s-c)}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{19(19-6)(19-14)(19-18)}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{18(13)(5)(1)}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle \sqrt{1170}\)

\(\,\,\,\,\,\,\text{Area}=\displaystyle 3\sqrt{130}\)

\(\text{The area is }\approx 35.14\) units squared

\(\textbf{7)}\) What is the area of this triangle?

Show Answer

The answer is \(\approx 5.2\) units squared

Show Work

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{(C)}\)

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}(4)(3) \sin{(60)}\)

\(\,\,\,\,\,\,\text{Area}=6 \cdot \frac{\sqrt{3}}{2}\)

\(\,\,\,\,\,\,\)The answer is \(\approx 5.2\) units squared

\(\textbf{8)}\) Find the area of this triangle.

Show Area

The area is \(= \displaystyle \frac{3\sqrt{3}}{4} \approx 1.2990 \) units squared

Show Work

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{(C)}\)

\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}(1)(3) \sin{(60)}\)

\(\,\,\,\,\,\,\text{Area}=\frac{3}{2} \cdot \frac{\sqrt{3}}{2}\)

\(\,\,\,\,\,\,\text{Area}= \displaystyle \frac{3\sqrt{3}}{4} \approx 1.2990\) units squared

See Related Pages\(\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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