Right triangle trigonometry
Knowing that a ladder leans at 70 degrees says nothing yet about how high it reaches. How can an angle be turned into lengths, and lengths back into an angle?
Where you are 7 inside
What this area is about
In a right triangle, nothing so far links a side to an angle. Similar triangles supply the link: every right triangle with the same acute angle has the same shape, so the ratio of any two sides is a number that belongs to the angle. Here the ratios are defined for an acute angle, between 0^\circ and 90^\circ; the unit circle later extends them to every angle.
How these fit together
Definitions. Opposite, adjacent and hypotenuse names each side relative to one acute angle. Sine, cosine and tangent are three ratios of those sides, and secant, cosecant and cotangent are their reciprocals.
Exact values. Trig values of the special right triangles reads exact values for 30^\circ, 45^\circ and 60^\circ off two familiar triangles.
Procedures. Solving right triangles goes from an angle and a side to the other sides. Inverse trigonometric ratios go from two sides back to an acute angle. Angles of elevation and depression apply both to heights and distances you cannot reach.
A path through it
Opposite, adjacent and hypotenuse first, then sine, cosine and tangent. After that come the reciprocal ratios, the special triangles, solving right triangles and inverse ratios; the inverse lesson refers back to solving right triangles, and elevation and depression build on solving right triangles.
Next: the unit circle carries these ratios to every angle, and the laws of sines and cosines carry them to triangles with no right angle.