Warm-UpDetermine whether the following triangles are acute, right or obtuse.

1. 7, 10, 13

2. 10, 8, 6

3. 4, 5, 6

Trigonometry Functions

Apply the Sine, Cosine and Tangent Ratios

VocabularyTrigonometric Ratio: A ratio of the lengths of two sides in a right triangle. Sine: A trigonometric ratio, abbreviated as sin.

Tangent: A trigonometric ratio, abbreviated as tan.

Cosine: A trigonometric ratio, abbreviated as cos.

VocabularyAngle of elevation: When you look up at an object, the angle that your line of sight makes with a line drawn horizontally.

Angle of depression: When you look down at an object, the angle that your line of sight makes with a line drawn horizontally.

Sine RatioSine Ratio:

Let ABC be a right triangle with acute A. The sine of A (written as sin A) is defined as follows:

Leg Opposite

A

A

sin A = length of leg opposite A length of the hypotenusehypotenuse

B

C

= BC AB

Cosine RatioCosine Ratio:

Let ABC be a right triangle with acute A. The cosine of A (written as cos A) is defined as follows:

Leg adjacent to AA

hypotenuse

B

C

cos A = length of leg adjacent to A length of the hypotenuse

= AC AB

Tangent RatioTangent Ratio:

Let ABC be a right triangle with acute A. The tangent of A (written as tan A) is defined as follows:

Leg adjacent to A

Leg Opposite

A

A

tan A = length of leg opposite A length of leg adjacent to Ahypotenuse

B

C

= BC AC

S O H C A H T O ASine

Opposi te

Opposi te

Hypotenuse

Hypotenuse

CoSine

Tangent

Adjacent

Adjacent

Example 1Find the value of x.

• Multiply both sides by x.

17°

9

x

x • tan 17° = 9 • Divide both sides by tan 17°

tan 17° = opp. adj. tan 17° = 9 x

x = 9 tan 17° x = 9 0.3057

= 29.44

Example 2Use the sine ratio to find the value of the variable. Round decimals to the nearest tenth.

sin A = opposite hypotenuse

14

25°

B

sin 25° = x 14

x = 5.9

C

x

A • Multiply both sides by 14. 14 • sin 25° = x

Example 3A rope, staked 20 feet from the base of a building, goes to the roof and forms an angle of 58° with the ground. To the nearest tenth of a foot, how long is the rope?

• Multiply both sides by x.

58°

x

x • cos 58° = 20 • Divide both sides by cos 58°

cos 58° = adj. hyp.

cos 58° = 20 x

x = 20 cos 58°

≈ 20 0.5299

≈ 37.7 ft

20 ft

Example 4Find the value of h.

• Multiply both sides by 24.

65°

h 24 • tan 65° = h • Use a calculator to simplify.

tan 65° = opp. adj. tan 65° = h 24

h = 51 feet 24

ft

HomeworkWorksheet