Trigonometry as angle-side relationships

Summary

Trigonometry is the study of relationships between angles and side lengths in triangles. For right triangles, the trigonometric functions (such as sine, cosine, and tangent) connect a chosen acute angle to ratios of the triangle’s sides. Because any two right triangles with the same acute angle are similar, these ratios depend only on the angle, not on the triangle’s overall size. Specifically, for an acute angle A in a right triangle: sin(A) is the ratio of the side opposite A to the hypotenuse, cos(A) is the ratio of the adjacent leg to the hypotenuse, and tan(A) is the ratio of the opposite leg to the adjacent leg. Their reciprocals define csc(A), sec(A), and cot(A). These angle-side relationships can be remembered using mnemonics like SOH-CAH-TOA, and they can also be visualized using the unit circle, where cos(A) and sin(A) appear as the x- and y-coordinates of the point where the terminal side of the angle intersects the circle.

Key Points

• 1

  Trigonometric functions relate an angle in a right triangle to ratios of its side lengths.

• 2

  sin(A)=opposite/hypotenuse, cos(A)=adjacent/hypotenuse, and tan(A)=opposite/adjacent.

• 3

  Reciprocal functions (csc, sec, cot) correspond to the inverses of sine, cosine, and tangent.

• 4

  Similarity of right triangles with the same acute angle makes these ratios depend only on the angle.

• 5

  The unit circle provides a geometric way to interpret sine and cosine as coordinates (x=cos A, y=sin A).