Check Problems from last lesson
The word trigonometry comes from Greek words that mean "triangle measurement."
When Egyptians first used a sundial around 1500 B.C., they were using trigonometry.
The picture to the right is a model of how a sundial works:
As the sun shines on the staff given by , it casts a shadow represented by . The staff is a fixed length, therefore the length of varies with the measure of angle , which changes as the sun moves throughout the day. The Egyptians looked at the ratio to determine the time. This ratio is called the tangent of .
A ratio of the lengths of two sides of a right triangle is called a tigonometric ratio. The three most common ratios are sine, cosine, and tangent. They are abbreviated sin,cos, and tan respectively.
Trigonometric Ratios Definition:
refer to the triangle to the right:
Trogonometric ratios are related to the acute angles of a triangle, not the right angle. The value of the trigonometric ratio, depends only on the angle. Consider the following three similar triangles (why are they similar?):
By the definition of the sin above we have , ,.
But since the triangles are similar, the ratio of their sides are equal, thus .
Activity: Open the GSP Sketch and use Calculate in the measure menu to calculate the sin,cos, and tangent of CAB by computing the ratios in the definition above.
Compare to the calculated sin,cos, and tangent.
Trigonometric ratios can be used to find the measures of acute angles in a right triangle when you know the measures of the two sides of the triangle. You must determine which ratio to use depending on which sides are given. Then, use the inverse trigonometric functions on your calculator to find the angle.
Example: Find the approximate measure of E.
We know the leg adjacent to E and the measure of the hypotenuse. We would therefore use the cosine, since it is defined in terms of the adjacent side and the hypontenuse:
Now on your calculator, make sure it is in degree mode and enter:
5 14 = INV COS
or on a TI-83 calculator: First hit mode key and select degree by using the arrows to put cursor over degree and pressing enter.
Now press 2nd COS 5 14 ) Enter
You should get 69.075168. So, E.
Click the lightbulb to practice what you've learned.