I know how to tell which one when you have two of the sides, but what if you only have one side and three angles? For example, what if you had to do the triangle
and the opposite was 80. How would you know if you had to use tan, cos, or sin to find a specific side?
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$\begingroup$Trig functions relate to right triangles. They apply to the other two angles that are not 90 degrees use the following definitions:
The terms opposing and adjacent applying to sides which are not the hypotenuse.
Then
$$ \begin{aligned} \sin ({\rm angle}) & = \frac{ {\rm opposing} }{ {\rm hypotenuse} } \\ \cos ({\rm angle}) & = \frac{ {\rm adjacent} }{ {\rm hypotenuse} } \\ \tan ({\rm angle}) & = \frac{ {\rm opposing} }{ {\rm adjacent} } \\ \end{aligned} $$
$\endgroup$ $\begingroup$We know that $\cos(\alpha)$ is defined by:$$\frac{a}{c}$$where $c$ is the hypotenuse and $a$ is the side adiacent to the angle $\alpha$. The $\sin(\alpha)$ is defined by:$$\frac{b}{c}$$where $b$ is the side opposite to the angle $\alpha$ and $c$ the hypotenuse. The $\tan(\alpha)$ is defined as the ratio between $\sin(\alpha)$ and $\cos(\alpha)$, so:$$\tan(\alpha)=\frac{b}{a}$$
In your case, you have first to calculate the lenght of $c$, using:$$c=\frac{b}{\sin(75°)}=\frac{80}{\sin(75°)}$$in fact, we know by what we have stated before that:$$80=c\cdot\sin(75°) \leftrightarrow c=\frac{80}{\sin(75°)}$$With this, you can find $a$, so:$$a=c\cdot \cos(\alpha)=80\cdot \tan(75°)$$
As @David G. Stork has pointed out, a mnemonical phrase you can learn is: "SOHCAHTOA (which reminds us of "sock it to me"): Sine = Opposite/ Hypotenuse; Cosine = Adjacent/ Hypotenuse; Tangent = Opposite/ Adjacent."
$\endgroup$ 8 $\begingroup$The simple answer is.... don't start with the trig function, start with the sides, then with that information, determine which trig ratio you need.
For example, suppose I know that the opposite side is 80, and I wish to find the hypotenuse. First off, build the fraction (and try to put the unknown in the denominator; it's just easier that way)$$\frac{\text{Opposite}}{\text{Hypotenuse}} $$
Now, look for the trig ratio that matches. $$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$Finally, just make substitutions for known quantities$$\sin 75^\circ = \frac{80}{x}$$
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