The Pythagorean Theorem and An Overview of Trigonometry
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Our goal is to find the two angles and lengths of all three sides of a right triangle. This is accomplished by using the Pythagorean Theorem and the three trigonometric functions sine, cosine, and tangent.
The side opposite an angle is the side not "touching" the angle. The hypotenuse is the word reserved for the side opposite the <Sign in to see all the formulas> angle. This is $r$ in the figure above. The side $a$ is opposite $\phi$ and $b$ is opposite <Sign in to see all the formulas> in the triangle above. The side adjacent to an angle is the side touching the angle that is not the hypotenuse. The side $b$ is adjacent to $\phi$ and $a$ is adjacent to <Sign in to see all the formulas>.
The Pythagorean Theorem gives a relationship between the three sides:
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Our three functions sine, cosine, and tangent give relationships between the sides and the angles:
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Putting this in terms of our diagram above we have:
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Let's work an example. Let's find the length of all the sides and all the angles in the right triangle below.
With <Sign in to see all the formulas> we can use sine or tangent to get one of the sides. We already know the other angle is $60^o$ because the sum of the angles in a triangle are $180^o$ so <Sign in to see all the formulas>.
We can find $g$ by using the sine function:
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Now to find the length $f$ by the Pythagorean Theorem:
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We have found the lengths of all sides and both angles! We can further verify all this by finding $f$ with the tangent function:
