4.03 Right Triangle Trigonometry

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Right Triangle Trigonometry

Consider a right triangle with one of the acute angles as the angle θ

The three sides are:

Hypotenuse
Side opposite the angle θ
Side adjacent to the angle θ

Right triangle with adjacent side, opposite side, and hypotenuse. Right angle between opposite side and adjacent side, theta angle between adjacent side and hypotenuse.

Right Triangle Trigonometry

The ratios of the lengths of these three sides are used to define the six trigonometric functions.

sin theta equals opposite side over hypotenuse, cos theta equals adjacent side over hypotenuse, tan theta equals opposite side over adjacent side

csc theta equals hypotenuse over opposite side, sec theta equals hypotenuse over adjacent side, cot theta equals adjacent side over opposite side

Abbreviations commonly used in right triangle trig: hypotenuse = hyp, opposite side = opp, adjacent side = adj

Finding the Trig Values

Find the exact values of the six trig functions for the angle θ

Right triangle with adjacent side equals 3, opposite side equals 4, and hypotenuse. Right angle between opposite side and adjacent side, theta angle between adjacent side and hypotenuse.

In order to find these trigonometric values, we need to know the lengths of all sides. We do not know the length of the hypotenuse, but can find it using the Pythagorean theorem.

Using the Pythagorean theorem, the length of the hypotenuse is:

h equals the square root of 3 squared plus 4 squared

h = √25

h = 5

Finding the Trig Values

Find the exact values of the six trig functions for the angle θ

Right triangle with adjacent side equals 3, opposite side equals 4, and hypotenuse equals 5. Right angle between opposite side and adjacent side, theta angle between adjacent side and hypotenuse.

Now that we know the lengths of all sides of the triangle, we can solve the ratios by plugging in the values for the appropriate sides.

Trig Functions 1-3

trig function sin theta = opp over hyp = blank over blank, trig function cos theta = adj over hyp = blank over blank, trig function tan theta = opp over adj = blank over blank

Answers:

trig function sin theta = opp over hyp = 4 over 5, trig function cos theta = adj over hyp = 3 over 5, trig function tan theta = opp over adj = 4 over 3

trig function csc theta = opp over hyp = blank over blank, trig function sec theta = adj over hyp = blank over blank, trig function cot theta = opp over adj = blank over blank

Answers:

trig function csc theta = opp over hyp = 5 over 4, trig function sec theta = adj over hyp = 5 over 3, trig function cot theta = opp over adj = 3 over 4

Find the exact values of the six trig functions for the angle θ. Review the solutions below.

Evaluate Trigonometric Functions of the Acute Angles of a Right Triangle

Open Evaluate Trigonometric Functions of the Acute Angles of a Right Triangle in a new tab

Frequently Used Angles 1

The angles 30° or , 45° or , and 60° or are used frequently in trigonometry. It is necessary to learn their exact values.

Construct a right triangle with one of the acute angles being 45° and the sides having a length of 1.

Right triangle with adjacent side equals 1, opposite side equals 1, and the hypotenuse. Right angle between opposite side and adjacent side, with both acute angles being 45 degrees.

Using the Pythagorean theorem, the length of the hypotenuse is:

h equals the square root of 1 squared plus 1 squared

h = √2

The angles 30° or , 45° or , and 60° or are used frequently in trigonometry. It is necessary to learn their exact values.

Right triangle with adjacent side equals 1, opposite side equals 1, and the hypotenuse is the square root of 2. Right angle between opposite side and adjacent side, with both acute angles being 45 degrees.

Now that we know the lengths of all sides of the triangle, we can solve the ratios by plugging in the values for the appropriate sides.

Trig Functions 1-6 Angles 1

sin theta = opp over hyp = blank over square root of 2, cos theta = adj over hyp = blank over square root of 2, tan theta = opp over adj = blank over 1, csc theta = hyp over opp = blank over 1, sec theta = hyp over adj = blank over 1, cot theta = adj over opp = blank over 1

Answers:

sin theta = opp over hyp = 1 over square root of 2, cos theta = adj over hyp = 1 over square root of 2, tan theta = opp over adj = 1 over 1, csc theta = hyp over opp = square root of 2 over 1, sec theta = hyp over adj = square root of 2 over 1, cot theta = adj over opp = 1 over 1

Simplify the fractions as necessary.

sine 45 degrees equals opp over hyp equals the fraction with numerator 1 and denominator the square root of 2 times the fraction with numerator the square root of 2 and denominator the square root of 2 equals the fraction with numerator the square root of 2 and denominator 2

cosine 45 degrees equals adj over hyp equals the fraction with numerator 1 and denominator the square root of 2 times the fraction with numerator the square root of 2 and denominator the square root of 2 equals the fraction with numerator the square root of 2 and denominator 2

tangent 45 degrees equals opp over hyp equals one over one equals 1

cosecant 45 degrees equals hyp over opp equals the fraction with numerator the square root of 2 and denominator 1 equals the square root of 2

secant 45 degrees equals hyp over adj equals the fraction with numerator the square root of 2 and denominator 1 equals the square root of 2

cotangent 45 degrees equals adj over opp equals one over one equals 1

Frequently Used Angles 2

Use the equilateral triangle shown to find the exact values of the trigonometric functions for 30° and 60°.

Use the Pythagorean theorem to find the length of the height of the triangle.

Let's start by finding the values for 30°. Now that we know the lengths of all sides of the triangle, we can solve the ratios by plugging in the values for the sides.

Trig Function 1-6 of 30° Angle

sin 30 degrees = opp over hyp = blank over 2, cos 30 degrees = adj over hyp = blank over 2, tan 30 degrees = opp over adj = blank over square root of 3, csc 30 degrees = hyp over opp = blank over 1, sec 30 degrees = hyp over adj = blank over square root of 3, cot 30 degrees = adj over opp = blank over 1

Answers:

sin 30 degrees = opp over hyp = one over 2, cos 30 degrees = adj over hyp = square root of 3 over 2, tan 30 degrees = opp over adj = 1 over square root of 3, csc 30 degrees = hyp over opp = 2 over 1, sec 30 degrees = hyp over adj = 2 over square root of 3, cot 30 degrees = adj over opp = square root of 3 over 1

At this point, we will simplify the values we found as needed to eliminate the square root terms from the denominator. To simplify a fraction with a radical in the denominator, you multiply both the numerator and the denominator by the radical.

sine 30 degrees equals opp over hyp equals the fraction with numerator 1 and denominator 2

cosine 30 degrees equals adj over hyp equals the fraction with numerator square root of 3 and denominator 2

tangent 30 degrees equals opp over adj equals the fraction with numerator 1 and denominator square root of 3 multiplied by square root of 3 over square root of 3 equals square root of 3 over 3

cosecant 30 degrees equals hyp over opp equals the fraction with numerator 2 and denominator 1 equals 2

secant 30 degrees equals hyp over adj equals the fraction with numerator 2 and denominator square root of 3 multiplied by square root of 3 over square root of 3 equals two square root of 3 over 3

cotangent 30 degrees equals adj over opp equals the fraction with numerator square root of 3 and denominator 1 equals square root of 3

Trig Function 1-6 of 60° Angle

Use the equilateral triangle shown to find the exact values of the trigonometric functions for 30° and 60°.

Now let's find the values for the 60° angle.

sin 60 degrees = opp over hyp = blank over 2, cos 60 degrees = adj over hyp = blank over 2, tan 60 degrees = opp over adj = blank over 1, csc 60 degrees = hyp over opp = blank over square root of 3, sec 60 degrees = hyp over adj = blank over 1, cot 60 degrees = adj over opp = blank over square root of 3

sine 60 degrees equals opp over hyp equals the fraction with numerator 1 and denominator 2

cosine 60 degrees equals opp over hyp equals the fraction with numerator 1 and denominator 2

tangent 60 degrees equals adj over opp equals the fraction with numerator square root of 3 and denominator 1 equals square root of 3

cosecant 60 degrees equals hyp over adj equals the fraction with numerator 2 and denominator square root of 3 multiplied by square root of 3 over square root of 3 equals 2 square root of 3 over 3

secant 60 degrees equals hyp over opp equals the fraction with numerator 2 and denominator 1 equals 2

cotangent 60 degrees equals opp over adj equals the fraction with numerator 1 and denominator square root of 3 multiplied by square root of 3 over square root of 3 equals square root of 3 over 3

Sines, Cosines, and Tangents of Special Angles
Degree	Radians	Sin θ	Cos θ	Tan θ
0°	0	0	1	0
30°
45°				1
60°
90°		1	0	undefined

Note:

sin 30° = = cos 60°

and

sin 60° = = cos 30°

Trigonometric Identities

From the preceding, the fundamental trigonometric identities can be defined by looking at the relationships between the functions and the three sides.