Right Triangle Trigonometry: Sine, Cosine, and Tangent Ratios. Solution of Problems.

Your thinking about trigonometry might be: What is the opposite side? Or What should I use in this problem? Sine, Cosine, or Tangent? What is the difference between sin 48° and sin-1(.7431)? How to solve for x, or how to find the side in the triangle? ... How do I read a table with sine, cosine and tangent ratios, and angles? How should I read a graph to find sine or cosine?

We really try to give you the easiest way of answering all those questions. The presentation of concepts and solution of problems privilege animations, sequence, and colors over long paragraphs difficult to understand. You will feel very supported all along the lesson!

Lesson's Content

Lesson's Glossary

Angle
Geometric shape formed by two rays (initial and ending sides of the angle) that share a common endpoint called the vertex. You may name an angle using the vertex, or a point in each ray and the vertex label in the center.

Angle of Elevation is the angle formed by the straight oblique line connecting a point in the horizontal and a point above the horizontal. As viewed by one observer for whose eye the horizontal is drawn.

Angle of depression is the angle formed by the straight oblique line connecting a point in the horizontal and a point below the horizontal. As viewed by one observer for whose eye the horizontal is drawn.

Cosine
Ratio determined by the hypotenuse in a right triangle and a side adjacent to a reference angle.

Hypotenuse
The side opposite the right angle in a right triangle.

Proportion
A statement that two ratios are equal.

Proportional
One of four numbers that form a true proportion.

Pythagorean theorem a2 + b2 = c2.
The Pythagorean theorem states that if you have a right triangle, then the square built on the hypotenuse is equal to the sum of the squares built on the other two sides.

Ratio
A quotient of 2 numbers.

Sine
Ratio of the length of the opposite side to a reference angle in a right triangle and the hypotenuse. Opposite/hypotenuse.

Tangent
The ratio of the length of the opposite side of a reference angle to the adjacent side to the same angle. Opposite over adjacent.

Interactive Geometric Applets: Relevant Theorems.

Sine is defined as opposite leg over hypotenuse; cosine is adjacent leg over hypotenuse;

and tangent is opposite leg over adjacent leg. This applet will allow you to verify these

relationships in a dynamic way. You will be able to drag any of the vertices in the

triangle to view how the values in the table are updated. It allows you to verify the

difference and similarity between a trigonometry ratio and its inverse.

Once you have found one leg, and the hypotenuse, or both legs you may use the Pythagorean

Theorem to find the missing side in the triangle. This applet will allow you to play with

any of the vertices, and verify how the theorem holds true. Observe that the red, and green

areas of the square at the legs fits in the square of the hypotenuse.

It is important to know that right triangle trigonometry works only for right triangles.

One way of figuring out if one triangle is right, when the sides have been given to you, it is

to use the converse of the Pythagorean Theorem. This states that given the three sides

of a triangle, this is right if applying the Pythagorean Theorem (taking the longest side as

the hypotenuse); you get a true equality statement. Shouldn't this be the case:

If the square of the longest side is greater than the sum of the squares of the other two,

then you have an obtuse triangle, and if it is less then it is an acute triangle.

Play with the below applet by dragging the point around and verifying in a visual,

and dynamic way this relationship.

In a triangle, when you have found two angles, then you may use the Angle Sum Theorem

to find the remaining angle. This applet allows you to drag any vertex, and verify that the

sum of the three angles is always 180°.

Vocabulary Puzzle Interactive

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