The National Council of Teachers of Mathematics recommends this activity for teaching the basics of trigonometry functions. On a piece of paper with 16 squares, write trigonometry expressions on each side of every square, 64 in total, making sure expressions on opposite sides of each side are equivalent values. Next, cut these squares along their sides and place the sixteen squares in an envelope. Instruct the students to match the equivalent expressions up again so that once each side of the small square has an equivalent value on the opposite side of the line.
The bases on a baseball diamond are a big square with each base 90 feet apart. A good activity to show how trigonometry is related to concepts in geometry, such as the Pythagorean Theorem, that students already know is to have them use new trigonometry concepts like sine and cosine to prove what they can also solve using tools they already know. First, have students calculate the length of the missing length of the right triangle between 2nd base and home base with the Pythagorean Theorem. Then have them use their new knowledge of cosine, sine, tangents and basic trigonometry functions to prove that the angles of the right triangle made by 2nd, 3rd, and home base are a 45-45-90 triangle.
Scout out a tree you know the approximate height of, and then measure 10 feet from its base. Ask yourself what the distance is between where you are standing and the top of the tree and at what angle is the incline? Using the law of tangents that the angle of any triangle is equal to the tangent of the opposite side (the height of the tree) and the adjacent side (10 feet), you can discover the angle of incline. Once this is known you can use the law of sine to find the length of the hypotenuse.
Using a ruler and a few stack books, have students stack three books on top of each other on top of a desk. Then ramp a ruler from the top of the books to the desk. A right triangle has been made. With a separate ruler measure the length along the desk from the edge of the books to where the ruler meets the desk. Since the length of the hypotenuse (the first ruler) and one of the sides is now known, use these measurement and sine, cosine, and tangent to determine the angle of declination.