Using Trig. Ratios to Find Distances

Using Trig. Ratios to Find Distances exercise

• Explain the law of cosines.

  Hints

  The first part of the law of cosines is the Pythagorean theorem.

  In the law of cosines, three sides and one angle are needed.

  The green sides as well as the green angle are known, while the orange side is unknown.

  Solution

  The law of cosines is a formula which can be used to find the unknown length of a side of a triangle, if you know the length of both other sides as well as the opposite angle of the unknown side.

  You can see the corresponding formulas in the picture to the right.

  How can you derive these formulas?

  The first part, $c^2=a^2+b^2$ (highlighted in green in the first equation in the picture), is the Pythagorean theorem. Then you still have to subtract a product: two times the product of the length of those sides on the right side of the equation and the cosine of the opposite angle.

  Let's try it with $a^2$ on the left-hand side:

  • We start with $a^2=b^2+c^2$, the Pythagorean property.
  • Next we subtract the double of the product of $b$ and $c$ as well as the cosine of the opposite angle of $a$, namely $A$, to get $a^2=b^2+c^2-2(bc)\cos(A)$.

• Determine the height of the tree house.

  Hints

  The law of cosines is given by, for example,

  $c^2=a^2+b^2-2ab\cos(C)$,

  where $C$ is the angle opposite to the side $c$.

  The wanted side squared is isolated on the left side of the equation. On the right side you only find known values.

  Remember PEMDAS, the order of calculation:

  • Paranthesis
  • Exponent
  • Multiplication and Division
  • Addition and Subtraction.

  Solution

  Because we know the lengths of two sides and we know the opposite angle of the side we would like to fine the length of, we can use the law of cosines.

  Let $a$ be the unknown height:

  $\begin{array}{rclll} a^2&=&~(15)^2+(22)^2\\ & &~-2((15)(22))(\cos(20.38°)) \end{array}$

  Using PEMDAS we can simplify as follows

  $\begin{array}{rclll} a^2&=&~225+484-660(0.9374)\\ &=&~90.31 \end{array}$

  Lastly, we take the square root to get

  $a\approx 9.5$.

  Thus the height of the wooden house is approximately $9.5$ ft.

• Calculate the height of the tree house.

  Hints

  $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

  is the law of sines.

  $a^2=b^2+c^2-2bc\cos A$

  is the law of cosines.

  For the law of sines two sides and two angles are needed, while for the law of cosines three sides and one angle are needed.

  The law of sines states that the ratio of any side and the sine of the opposite angle is the same as the ratio of any of the other sides and the sine of its opposite angle.

  Solution

  How can we decide, which law we have to use?

  Just look at the given information: here we know two angles (and thus three angles) and only one side. For the law of cosines we need at least two sides. So we use the law of sines.

  First we determine the last angle using the facts that the three corners sum to $180$°:

  $180°-64°-39°=77°$.

  To get the desired side $h$, we do as follows using the law of sines:

  $\frac{h}{\sin(39°)}=\frac{5.3}{\sin(77°)}$.

  Almost done, we still have to multiply by $\sin(39°)$ to get

  $h=\frac{5.3}{\sin(77°)}\times \sin(39°)\approx 3.4$.

  So we can conclude the height of the porch is given by $h\approx 3.4$ ft.

• Figure out the length of the side $x$.

  Hints

  Here you see the law of sines.

  Here you see the law of cosines.

  Solution

  Let's start with the triangle on the left and proceed to the right.

  • The following values are known: two angles, $30$° and $70$°, and the side of length $10$ opposite the angle of $30$°. We use the law of cosine: $\frac{10}{\sin(30°)}=\frac{x}{\sin(70°)}$. Multiplying both sides by $\sin(70°)$ gives us $x=\frac{10}{\sin(30°)}\times \sin(70°)\approx 18.8$.
  • We know the angle $75$° opposite to the unknown side and the two other sides, $7$ and $5$. So we use the law of cosines: $x^2=7^2+5^2-2(7)(5)\cos(75°)$. Using PEMDAS we get $x=49+25-70(0.2588)=55.9$ and taking the square root leads to $x\approx7.5$.
  • Here we know two angles, $65°$ and $72°$, and the side of length $12$ opposite the angle $65$°. We use the law of cosine: $\frac{12}{\sin(65°)}=\frac{x}{\sin(72°)}$. Multiplying with $\sin(72°)$ gives us $x=\frac{12}{\sin(65°)}\times \sin(72°)\approx 12.6$.
  • For the last triangle we use the law of cosines again. We know the angle $62$° opposite to the unknown side and the two other sides, $8$ and $3$. $x^2=8^2+3^2-2(8)(3)\cos(62°)$. Using PEMDAS we get $x=64+9-48(0.4694)=50.5$ and taking the square root leads to $x\approx7.1$.

• Clarify the law of sines.

  Hints

  If you know two angles and the length of one of the sides opposite one of these angles, then you can determine the length of the side opposite to the other angle.

  For the given pictured situation you can conclude

  $\frac{15}{\sin(96.77°)}=\frac{c}{\sin(33.23°)}$.

  Each time the side is written in the numerator.

  Solution

  The law of sines states that the ratio of any side and the sine of the opposite angle is the same as the ratio of any of the other side and the sine of its opposite angle.

  We can write it is as follows

  $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$.

  Pictured beside we have an example; the law of sines tells us that

  $\frac{15}{\sin(96.77°)}=\frac{c}{\sin(33.23°)}$.

• Examine the height of the tower.

  Hints

  The law of cosines is given by

  $a^2=b^2+c^2-2bc\cos(A)$,

  where $A$ is the angle opposite to the side $a$.

  Pay attention to PEMDAS; first multiply or divide, then add or subtract.

  You first have to calculate $2bc\cos(A)$ and then subtract the resulting product after.

  Don't forget to take the square root.

  Solution

  All we know is given in the picture beside.

  We know two sides, $420$ yds and $400$ yds, as well as one angle, $3$°. The side opposite to this angle is wanted. Thus we take the law of cosines:

  $~$

  $\begin{array}{rclll} x^2&=&~(420)^2+(400)^2\\ & &~-2((420)(400))(\cos(3°)) \end{array}$

  $~$

  Using PEMDAS we can simplify as follows:

  $\begin{array}{rclll} x^2&=&~176400+160000-336000(0.9986)\\ &=&~860.5 \end{array}$

  $~$

  Last we take the square root to get $a\approx 29$.

  So we can conclude the height of the tower by $29$ yds.