6.5 Sine, Cosine and Tangent Ratios and Applications of Trigonometry

We labelled the sides with a lowercase letter to match the uppercase letter of the opposite vertex.

[latex]c[/latex] is hypotenuse
[latex]a[/latex] is opposite
[latex]b[/latex] is adjacent

[latex]\begin{align*}sin(\theta) &=\frac{f}{d}\\[2ex]cos(\theta)&=\frac{e}{d}\\[2ex]tan(\theta)&=\frac{f}{e}\end{align*}[/latex]

Step 1: Read problem and make sure all the words and ideas are understood. Draw the right triangle and label  the given parts.

A drawing is given. Angle [latex]S[/latex] is our reference angle, [latex]R[/latex] is opposite side, [latex]r=4[/latex] is the adjacent side, and [latex]P[/latex] is the hypotenuse.

Step 2: Identify what you are looking for.

The hypotenuse.

Step 3: Label what we are looking for by choosing a variable to represent it.

[latex]\begin{eqnarray*}p&=&?\end{eqnarray*}[/latex]

Step 4: Find the required trigonometric ratio.

[latex]\begin{eqnarray*}cos(32^\circ)&=&\frac{4}{p}\end{eqnarray*}[/latex]

Step 5: Solve the ratio using good algebra techniques.

[latex]\begin{eqnarray*}0.8480&=&\frac{4}{p}\\p&=&4.7170\;\;\text{Rounding the ratios to 4 decimal places}\end{eqnarray*}[/latex]

Step 6: Check the answer in the problem and by making sure it makes sense.

[latex]\begin{eqnarray*}0.8480&\overset?=&\frac{4}{4.7170}\\0.8480&=&0.8480\checkmark\end{eqnarray*}[/latex]

Step 7: Answer the question with a complete sentence.

The hypotenuse is [latex]4.7[/latex]. Round final answer to one decimal place.

Use your calculator and press the 2nd FUNCTION key and then press the SIN, COS, or TAN key

a.  [latex]\begin{align*}A&=sin^{-1}(0.5)\\\angle{A}&=30^\circ\end{align*}[/latex]

b.  [latex]\begin{align*}B&=cos^{-1}(0.9735)\\\angle{B}&=13.2^\circ\end{align*}[/latex]    Rounded to one decimal place

c. [latex]\begin{align*}C&=tan^{-1}(2.89358)\\\angle{C}&=70.9^\circ\end{align*}[/latex]    Rounded to one decimal place

Step 1: Read problem and make sure all the words and ideas are understood. Draw the right triangle and label  the given parts.

A drawing is given. Angle [latex]T[/latex] is our reference angle,  [latex]t=7[/latex] is the opposite side,  [latex]s[/latex] is adjacent side, and [latex]r=11[/latex] is the hypotenuse.

Step 2: Identify what you are looking for.

Angle [latex]T[/latex].

Step 3: Label what we are looking for by choosing a variable to represent it.

[latex]\begin{eqnarray*}\angle T&=&?\end{eqnarray*}[/latex]

Step 4: Find the required trigonometric ratio.

[latex]\begin{eqnarray*}\sin(T)\;&=&\;\frac7{11}\end{eqnarray*}[/latex]

Step 5: Solve the ratio using good algebra techniques.

[latex]\begin{eqnarray*}sin(T)&=&0.6364\\T&=&sin^{-1}(0.6364)\\\angle T&=&39.5239^\circ\end{eqnarray*}[/latex]

Step 6: Check the answer in the problem and by making sure it makes sense.

[latex]\begin{align*}sin\left(39.523^\circ\right)&\overset?=0.6364\\0.6364&=0.6364\checkmark\end{align*}[/latex]

Step 7: Answer the question with a complete sentence.

The missing angle [latex]T[/latex] is [latex]39.5^\circ[/latex].

Step 1: Read problem and make sure all the words and ideas are understood. Draw the right triangle and label  the given parts.

A drawing is given. Angle [latex]A[/latex] is our reference angle,  [latex]a=9[/latex] is the opposite side,  [latex]c=5[/latex]  is the adjacent side, and [latex]b[/latex] is the hypotenuse.

Step 2: Identify what you are looking for.

Angle [latex]A[/latex].

Step 3: Label what we are looking for by choosing a variable to represent it.

[latex]\begin{eqnarray*}\angle A&=&?\end{eqnarray*}[/latex]

Step 4: Find the required trigonometric ratio.

[latex]\begin{eqnarray*}tan A&=&\frac{9}{5}\end{eqnarray*}[/latex]

Step 5: Solve the ratio using good algebra techniques.

[latex]\begin{eqnarray*}tan(A)&=&1.8\\A&=&tan^{-1}(1.8)\\\angle A&=&60.9^\circ\end{eqnarray*}[/latex]

Step 6: Check the answer in the problem and by making sure it makes sense.

[latex]\begin{eqnarray*}tan\;60.9^\circ&\overset?=&\;1.8\\1.8&=&1.8 \checkmark\end{eqnarray*}[/latex]

Step 7: Answer the question with a complete sentence.

The missing angle [latex]A[/latex] is [latex]60.9^\circ[/latex].

Since the sum of angles in any triangle is [latex]180^\circ[/latex], the measure of angle B can be easy calculated.

[latex]\begin{align*}\angle B&=180^\circ-90^\circ-42^\circ\\\angle B&=48^\circ\end{align*}[/latex]

Step 1: Read problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.

A drawing is given. Angle [latex]A[/latex] is our reference angle, [latex]a=8[/latex] is the opposite side, [latex]b[/latex] is the adjacent side, and [latex]c[/latex] is the hypotenuse.

Step 2: Identify what you are looking for.

a. The adjacent side.
b. The hypotenuse.

Step 3: Label what we are looking for by choosing a variable to represent it.

[latex]\begin{eqnarray*}\text{a.}\;b&=&?\\ \text{b.}\;c&=&?\end{eqnarray*}[/latex]

Step 4: Find the required trigonometric ratio.

[latex]\begin{eqnarray*}\text{a.}\;tan(42^\circ)&=&\frac{8}{b}\\[2ex] \text{b.}\;sin(42^\circ)&=&\frac{8}{c}\end{eqnarray*}[/latex]

Step 5: Solve the ratio using good algebra techniques.

[latex]\begin{eqnarray*}\text{a.}\;0.9004&=&\frac{8}{b}\\0.9004 b&=&8\\b&=&8.8849\\[3ex]\text{b.}\;0.6691&=&\frac{8}{c}\\0.6691 c&=&8\\c&=&11.9563\end{eqnarray*}[/latex]

Step 6: Check the answer in the problem and by making sure it makes sense.

[latex]\begin{eqnarray*}\text{a.}\;tan(42^\circ)&\overset?=&\frac8{8.8849}\\0.9&=&0.9 \checkmark\\[3ex]\text{b.}\;sin(42^\circ)&\overset?=&\frac8{11.9563}\\0.6691&=&0.6691 \checkmark\end{eqnarray*}[/latex]

Step 7: Answer the question with a complete sentence.

a. The adjacent side is [latex]8.9[/latex]. Rounded to one decimal place.
b. The hypotenuse is [latex]12[/latex].

We solved the right triangle

[latex]\begin{align*}\angle A&=42^\circ\\\angle B&=48^\circ\\\angle C&=90^\circ\\[2ex]a&=8\\b&=8.9\\c&=12\end{align*}[/latex]

Step 1: Read problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.

A drawing is given. Let angle [latex]D[/latex] be our reference angle,  [latex]d=4[/latex] is the opposite side,  [latex]f[/latex] is the adjacent side, and [latex]e=9[/latex] is the hypotenuse.

Step 2: Identify what you are looking for.

a. Angle D.
b. The adjacent.

Step 3: Label what we are looking for by choosing a variable to represent it.

[latex]\begin{eqnarray*}\text{a.}\;\angle D&=&?\\[2ex] \text{b.}\;f&=&?\end{eqnarray*}[/latex]

Step 4: Find the required trigonometric ratio.

[latex]\begin{eqnarray*}\text{a.}\;sin(D)&=&\frac{4}{9}\\[2ex] \text{b.}\;4^2+f^2&=&9^2\end{eqnarray*}[/latex]

Step 5: Solve the ratio using good algebra techniques.

[latex]\begin{align*} &\text{a.}&sin(D)&=0.4444\\ &\;&D&=sin{-1}{0.4444}\\ &\;&\angle D&=26.3850^\circ\\[3ex] &\text{b.}&16+f^2&=81\\ &\;&f^2&=81-16\\ &\;&f^2&=65\\ &\;&f&=\sqrt{65}\\ &\;&f&=8.06 \end{align*}[/latex]

Step 6: Check the answer in the problem and by making sure it makes sense.

[latex]\begin{align*} &\text{a.}&sin(26.3850^\circ)&\overset?=\frac49\\ &\;&0.4444&=0.4444\checkmark\\[3ex] &\text{b.}&4^2+8.06^2&\overset?=9^2\\ &\;&81&=81\checkmark \end{align*}[/latex]

Step 7: Answer the question with a complete sentence.

a. The missing angle [latex]D[/latex] is [latex]26.39^\circ[/latex].
b. The adjacent side is [latex]8.06[/latex]. Rounded to two decimal places.

The missing angle [latex]\begin{eqnarray*}F&=&180^\circ - 90^\circ - 26.39^\circ\\F&=&63.61^\circ\end{eqnarray*}[/latex].

We solved the right triangle

[latex]\begin{align*}\angle D\ &=26.39^\circ\\\angle E\ &=90^\circ\\\angle F\ &=63.61^\circ\\[2ex]d &=4\\e &=9\\f &=8.06\end{align*}[/latex]