8.6: Sine, Cosine, Tangent
What if you were given a right triangle and told that its sides measure 3, 4, and 5 inches? How could you find the sine, cosine, and tangent of one of the triangle's nonright angles? After completing this Concept, you'll be able to solve for these trigonometric ratios.
Watch This
CK12 Foundation: Chapter8SoneCosineTangentA
Watch the parts of the video dealing with the sine, cosine, and tangent.
James Sousa: Introduction to Trigonometric Functions Using Triangles
Guidance
The word trigonometry comes from two words meaning triangle and measure. In this lesson we will define three trigonometric (or trig) functions.
Trigonometry: The study of the relationships between the sides and angles of right triangles.
In trigonometry, sides are named in reference to a particular angle. The hypotenuse of a triangle is always the same, but the terms adjacent and opposite depend on which angle you are referencing. A side adjacent to an angle is the leg of the triangle that helps form the angle. A side opposite to an angle is the leg of the triangle that does not help form the angle. We never reference the right angle when referring to trig ratios.
The three basic trig ratios are called, sine, cosine and tangent. At this point, we will only take the sine, cosine and tangent of acute angles. However, you will learn that you can use these ratios with obtuse angles as well.
Sine Ratio: For an acute angle
Cosine Ratio: For an acute angle
Tangent Ratio: For an acute angle
There are a few important things to note about the way we write these ratios. First, keep in mind that the abbreviations
A few important points:
 Always reduce ratios when you can.
 Use the Pythagorean Theorem to find the missing side (if there is one).
 The tangent ratio can be bigger than 1 (the other two cannot).
 If two right triangles are similar, then their sine, cosine, and tangent ratios will be the same (because they will reduce to the same ratio).
 If there is a radical in the denominator, rationalize the denominator.
 The sine, cosine and tangent for an angle are fixed.
Example A
Find the sine, cosine and tangent ratios of
First, we need to use the Pythagorean Theorem to find the length of the hypotenuse.
So,
Example B
Find the sine, cosine, and tangent of
Find the length of the missing side.
Therefore,
Example C
Find the sine, cosine and tangent of
This is a special right triangle, a 306090 triangle. So, if the short leg is 6, then the long leg is
Watch this video for help with the Examples above.
CK12 Foundation: Chapter8SineCosineTangentB
Concept Problem Revisited
The trigonometric ratios for the nonright angles in the triangle above are:
Vocabulary
Trigonometry is the study of the relationships between the sides and angles of right triangles. The legs are called adjacent or opposite depending on which acute angle is being used. The three trigonometric (or trig) ratios are sine, cosine, and tangent.
Guided Practice
Answer the questions about the following image. Reduce all fractions.
1. What is
2. What is
3. What is
Answers:
1.
2.
3.
Practice
Use the diagram to fill in the blanks below.

tanD=?? 
sinF=?? 
tanF=?? 
cosF=?? 
sinD=?? 
cosD=??
From questions 16, we can conclude the following. Fill in the blanks.

cos−−−−=sinF andsin−−−−=cosF . 
tanD andtanF are _________ of each other.
Find the sine, cosine and tangent of
 Explain why the sine of an angle will never be greater than 1.
 Explain why the tangent of a \begin{align*}45^\circ\end{align*}
45∘ angle will always be 1.  As the degree of an angle increases, will the tangent of the angle increase or decrease? Explain.
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Acute Angle
An acute angle is an angle with a measure of less than 90 degrees.Adjacent Angles
Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'nextto'.Hypotenuse
The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.Legs of a Right Triangle
The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.opposite
The opposite of a number is . A number and its opposite always sum to zero.Pythagorean Theorem
The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by , where and are legs of the triangle and is the hypotenuse of the triangle.Radical
The , or square root, sign.sine
The sine of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the hypotenuse.Trigonometric Ratios
Ratios that help us to understand the relationships between sides and angles of right triangles.Image Attributions
Here you'll learn what the three trigonometric ratios are and how to find their value for a right triangle's nonright angle.