The Primary Trigonometric Ratios

In a right triangle, the side across from the right angle is the hypotenuse. It is the longest side of the triangle. The other two sides, the legs, are named from the point of view of one of the acute angles, called the reference angle (often written $\theta$, the Greek letter "theta"). The opposite side is the leg across from the reference angle; it does not touch that angle. The adjacent side is the leg that forms one arm of the reference angle (the hypotenuse is the other arm). The hypotenuse stays the same whichever acute angle you use, but opposite and adjacent swap when you switch reference angles. A common way to name sides: in $△ABC$, the lowercase letter names the side opposite the vertex with the same capital letter, so $a=BC$ is opposite $\angle A$, $b=AC$ is opposite $\angle B$, and $c=AB$ is opposite $\angle C$.

Take any two right triangles that each have a $35^{\circ }$ angle. They have a right angle in common and a $35^{\circ }$ angle in common, so they are similar by AA. If the larger one is the smaller one multiplied by a scale factor $k$, then its opposite side is $k\times \text{opp}$ and its hypotenuse is $k\times \text{hyp}$. Dividing, the $k$ cancels: $\frac{k\times \text{opp}}{k\times \text{hyp}}=\frac{\text{opp}}{\text{hyp}}$. So the ratio of two sides within one right triangle equals the ratio of the two corresponding sides in any similar right triangle. The table shows three right triangles with a $35^{\circ }$ angle, measured to three decimal places: the sizes double each time, but the ratios do not change.

Your calculator gives these ratios for any angle, but it must be in degree mode (look for DEG on the screen). A quick test: $\sin 30^{\circ }$ should be exactly $0.5$. If you get about $-0.988$, the calculator is in radian mode and every answer will be wrong. Trigonometric ratios are usually rounded to four decimal places when written down. Notice the patterns in the table below. For an acute angle, $\sin \theta$ and $\cos \theta$ are always between $0$ and $1$, because each leg is shorter than the hypotenuse. As the angle grows from $0^{\circ }$ toward $90^{\circ }$, the sine grows and the cosine shrinks; $\sin \theta =\cos (90^{\circ }-\theta )$, because the side opposite one acute angle is adjacent to the other. The tangent can be any positive number: it is less than $1$ below $45^{\circ }$, exactly $1$ at $45^{\circ }$ (the two legs are equal), and greater than $1$ above $45^{\circ }$.

To find a side, label the sides from the given acute angle, then pick the ratio that uses the side you know and the side you want. If the unknown is on the top of the ratio, multiply: in a right triangle with a $28^{\circ }$ angle and a $15.0$ cm hypotenuse, the opposite side $x$ satisfies $\sin 28^{\circ }=\frac{x}{15.0}$, so $x=15.0\sin 28^{\circ }\approx 7.0$ cm. If the unknown is on the bottom, it moves when you solve: $\cos 52^{\circ }=\frac{8.0}{x}$ becomes $x\cos 52^{\circ }=8.0$, so $x=\frac{8.0}{\cos 52^{\circ }}\approx 13.0$ cm. A useful check: the hypotenuse must come out longer than both legs.

To find an angle, you know two sides and want the angle. Choose the ratio that uses those two sides, work it out as a number, then use the inverse ratio on your calculator (usually SHIFT or 2nd, then the sin, cos or tan key). If $\tan A=\frac{7.0}{9.5}$, then $A={\tan }^{-1}\left(\frac{7.0}{9.5}\right)\approx 36^{\circ }$. Read ${\tan }^{-1}$ as "the angle whose tangent is". The $-1$ here is not an exponent: ${\sin }^{-1}(0.5)=30^{\circ }$ is an angle (the acute angle whose sine is $0.5$), while $\frac{1}{\sin 30^{\circ }}=2$ is just a number. Once one acute angle is known, the other is $90^{\circ }$ minus it, because the three angles add to $180^{\circ }$ and the right angle uses $90^{\circ }$ of that.

Example: Trig to solve the sides and angles of a right triangle | Trigonometry | Khan Academy