Deck 11: Introduction to Trigonometry

Question 1

For the right triangle with right , prove that .

Answer 1

By definition, , , and . Then , which can also be written . Then . Thus, .

Question 2

For the right triangle with right , prove that . Note that and are the same and that as shown in ; also, and have the
same meaning.

Answer 2

With and , we see that or .
Then . By using the Pythagorean Theorem, we know that . It follows that , so .

Question 3

Use the drawings provided to prove the following theorem.
"The area of an acute triangle equals one-half the product of the lengths of two sides of a triangle and the sine of the included angle."
Given: Acute Prove:

Answer 3

The area of is given by . Considering the auxiliary altitude from vertex B
to side , we see that is a right angle. Then in right triangle .
From , it follows that . By substitution, the area formula becomes so we have .

Question 4

For the right triangle with right , prove that .