TRIGONOMETRYLesson 1: Primary Trigonometric Ratios

Todays Objectives

• Students will be able to develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles, including:• Identify the hypotenuse of a right triangle and the

opposite and adjacent sides for a given acute angle in the triangle

• Explain the relationships between similar right triangles and the definitions of the primary trigonometric ratios

Right Triangles• A right triangle has a right (90º) angle. The other two

angles are acute (between 0º and 90º). • The side opposite the right angle is the longest side, and

is called the hypotenuse. • The two sides adjacent to the right angle are called legs of

the right triangle. The word adjacent means “beside”.

Right Angle

LegsAcute Angles

Hypotenuse

Pythagorean Theorem• In any right triangle, the square of the hypotenuse is equal to the

sum of the squares of the other two sides (legs). This can be illustrated as follows:

• Note: Vertices are often labeled with capital letters, and the sides opposite the vertices are labeled with the corresponding lower case letters. We normally label the hypotenuse as c.

C

bc

a

A

B

Example• Solve for the unknown side length, x. (a = 5, b = x, c = 11)

• Solution: Apply the pythagorean theorem,

5 cm11 cm

x

Sum of the Angles in a Triangle• In any triangle, the sum of the measures of the three

angles is always equal to 180º

A

B

C45°

100°

35°

Example • Determine the measure of angle XYZ. (When the name of

the angle is given as three letters, the middle letter represents the vertex of the angle)

• Solution: 180°- 85° - 47° = 48°

Y

X

Z

85°

47°

Sine, Cosine, and Tangent Ratios• The three primary trigonometric ratios describe the ratios

of the different sides in a right triangle.• These ratios use one of the acute angles as a point of

reference. The 90º angle is never used. In the following illustration, the ratios are described relative to angle θ.

• Notice that the abbreviations for sine, cosine, and tangent are sin, cos, and tan.

hypotenuse

Opposite side to θ

Adjacent side to θ

θ

SOHCAHTOA

• SOHCAHTOA• You can use the acronym SOH-CAH-TOA to remember

these ratios

Example• Determine the three primary trigonometric ratios from

angle θ

• Solution: first, find the unknown side x • = =

• Now, write the three ratios

θ 13 cm

12 cm

x

Trigonometric Ratios and Similar Triangles

• Similar triangles are triangles in which the corresponding angles have the same measure. The corresponding sides in similar triangles are proportional. One way of constructing similar right triangles is shown in the given diagram below.

B3

B2

B1

AC2C3 C1

35 cm

26 cm

29 cm

37 cm14 cm25 cm

Similar Triangles• Angles with the same markings have the same measure.

Three similar triangles have been formed: ∆AB1C1, ∆AB2C2, ∆AB3C3

• Using each of these triangles, consider the sine ratio for angle A. Because the sides are proportional, the sine ratios using each of the three similar triangles are equal.

Example• 1) State the tangent of angle θ using the labels of the sides• 2) Use a metric ruler to measure the side lengths for each triangle, and

give an estimate of the value of tan θ to the nearest hundredth• 3) Calculate the value of angle θ

• Note: Calculators need to be set to Degree Mode, not Radian Mode. Inverse functions are available by using the shift or 2nd function key on your calculator.

B3

B2

B1

AC3 C2 C1