Week 17 – Trigonometry – Manroop's Blog

This week in Pre-Calculus 11 we started the trigonometry unit. This week we learned about the sine law and the cosine law.

What is the Sine Law? The sine law is typically used when you are not dealing with right triangles. Although the sine law could be used with right triangles it involves more steps. When solving a triangle that doesn’t have a right angle you cannot use SOH CAH TOA, you have to use the sine law or cosine law. The formula for finding an angle is $\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{sin(C)}{c}$ . The formula for finding a side length is the reciprocal $\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$ .

Ex. In a triangle $\angle(B)=34$ , side $c$ is 14.0cm and $\angle(C)=65$ . What is the side length of $b$ to the nearest tenth?

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

$\frac{a}{\sin(A)}=\frac{b}{\sin(34)}=\frac{14}{\sin(65)}$

$\frac{b}{\sin(34)}=\frac{14}{\sin(65)}$

$b=\frac{14\times\sin(34)}{\sin65}$

$b=\frac{7.82}{\sin65}$

$b=8.638$

$b=8.6cm$

In the example above, the first step I did was plug in the known values into the formula. I knew this triangle was not a right triangle because it does not have a $\angle90$ . The second step was to isolate the variable and find it’s value.

Cosine Law: The Cosine Law is used when the Sine Law cannot be used. The cosine law can calculate a missing side length or angle. The formula for finding a missing side length is $a^2=b^2+c^2-2bc\cos(A)$ . The formula for finding an angle is $\cos(A)=\frac{b^2+c^2-a^2}{2bc}$ .

Ex. $\triangle{MKT}$ , side m is 12 cm long, side k is 7 cm and side t is 13 cm long. What is $\angle{K}$ to the nearest degree?

$\cos{K}=\frac{m^2+t^2-k^2}{2(mt})$

$\cos{K}=\frac{12^2+13^2-7^2}{2(12)(13)}$

$\cos{K}=\frac{144+169-49}{312}$

$\cos{K}=\frac{264}{312}$

$\cos{K}=0.846$

$K=\cos^{-1}{0.846}$

$K=32.2$

$K=32$ degrees

In the example above I used the cosine law to solve for $\angle{K}$ . The first step was to fill in the values. After that, I solved for $\angle{K}$ by calculating one side of the equation and then using the inverse function of cosine to solve for $\angle{K}$

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25	26	27	28	29	30

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