Chapter 7-5 Notes

Definitions

• Trigonometry

• The study of the relationships between the sides and of right

triangles

• Sides are named in reference to a particular

• Adjacent

• A side of a triangle that helps form the angle in question

• Opposite

• A side of a triangle that does not help form the angle in question

angles

angle

Examples

• Example 1

Which side is adjacent to A? Which side is opposite B?

b b

Definitions

• Sine Ratio

• For an acute angle x in a right triangle, the is equal to the ratio of the

side the angle over the of the triangle.

• Using the triangle in example 1: and

sin(x)

opposite hypotenuse

𝒔𝒊𝒏 𝑨 =𝒂

𝒄 𝒔𝒊𝒏 𝑩 =𝒃

𝒄

Definitions

• Cosine Ratio

• For an acute angle x in a right triangle, the is equal to the ratio of the

side to the angle over the of the triangle.

• Using the triangle in Example 1: and

cos(x)

adjacent hypotenuse

𝒄𝒐𝒔 𝑨 =𝒃

𝒄𝒄𝒐𝒔 𝑩 =

𝒂

𝒄

Definitions

• Tangent Ratio

• For an acute angle x in a right triangle, the is equal to the ratio of the

side the angle over the side to the angle.

• Using the triangle from Example 1: and

tan(x)

opposite adjacent

𝒕𝒂𝒏 𝑨 =𝒂

𝒃 𝒕𝒂𝒏 𝑩 =𝒃

𝒂

Note

• An easy way to remember the Trig Ratios is SOHCAHTOA

Examples

• Example 2

Find the sine, cosine, and tangent ratios of A.

𝒔𝒊𝒏 𝑨 =𝟏𝟐

𝟏𝟑𝟓𝟐 + 𝟏𝟐𝟐 = 𝒙𝟐

𝟐𝟓 + 𝟏𝟒𝟒 = 𝒙𝟐

𝟏𝟔𝟗 = 𝒙𝟐

𝟏𝟑 = 𝒙

𝒄𝒐𝒔 𝑨 =𝟓

𝟏𝟑

𝒕𝒂𝒏 𝑨 =𝟏𝟐

𝟓

Note

• Always reduce when you can.

• Use to find the missing side.

• The tangent ratio bigger than 1.

• If two right triangles are similar, then their sine, cosine, and tangent

ratios will be

• If there is a radical in the denominator,

ratios

Pythagorean Theorem

can be

the same

rationalize it

Examples

• Example 3

Find the sine, cosine, and tangent of B.

𝟓𝟐 + 𝒙𝟐 = 𝟏𝟓𝟐

𝟐𝟓 + 𝒙𝟐 = 𝟐𝟐𝟓

𝒙𝟐 = 𝟐𝟎𝟎

𝒙 = 𝟏𝟎 𝟐

𝒔𝒊𝒏 𝑩 =𝟏𝟎 𝟐

𝟏𝟓=

𝟐 𝟐

𝟑

𝒄𝒐𝒔 𝑩 =𝟓

𝟏𝟓=

𝟏

𝟑

𝒕𝒂𝒏 𝑩 =𝟏𝟎 𝟐

𝟓= 𝟐 𝟐

Examples

• Example 4

Find the sine, cosine, and tangent of 30°.

𝒙 = 𝟐(𝟔)𝒙 = 𝟏𝟐

𝒚 = 𝟔 𝟑

𝒔𝒊𝒏 𝟑𝟎 =𝟔

𝟏𝟐=

𝟏

𝟐

𝒄𝒐𝒔 𝟑𝟎 =𝟔 𝟑

𝟏𝟐 =𝟑

𝟐

𝒕𝒂𝒏 𝟑𝟎 =𝟔

𝟔 𝟑=

𝟑

𝟑

Note

• The sine, cosine, and tangent values for an angle are fixed

Examples

• Example 5

Find the trig value using your calculator:

sin(78) cos(60) tan(15)

𝒔𝒊𝒏 𝟕𝟖 = 𝟎. 𝟗𝟕𝟖 𝒄𝒐𝒔 𝟔𝟎 = 𝟎. 𝟓 𝒕𝒂𝒏 𝟕𝟖 = 𝟎. 𝟐𝟔𝟖

Examples

• Example 6

Find the value of each variable. Round your answer to the nearest hundredth.

𝒄𝒐𝒔 𝟐𝟐 =𝒂

𝟑𝟎

𝟑𝟎 ∗ 𝒄𝒐𝒔 𝟐𝟐 = 𝒂

𝟐𝟕. 𝟖𝟐 = 𝒂

𝒔𝒊𝒏 𝟐𝟐 =𝒃

𝟑𝟎

𝟑𝟎 ∗ 𝒔𝒊𝒏 𝟐𝟐 = 𝒃

𝟏𝟏. 𝟐𝟒 = 𝒃

Examples

• Example 7

Find the value of each variable. Round your answer to the nearest hundredth.

𝒄𝒐𝒔 𝟒𝟐 =𝟗

𝒄

𝒄 ∗ 𝒄𝒐𝒔 𝟒𝟐 = 𝟗

𝒄 =𝟗

𝒄𝒐𝒔(𝟒𝟐)

𝒕𝒂𝒏 𝟒𝟐 =𝒅

𝟗

𝟗 ∗ 𝒕𝒂𝒏 𝟒𝟐 = 𝒅

𝟖. 𝟏 = 𝒅

𝒄 = 𝟏𝟐. 𝟏𝟏

Definitions

• Angle of Depression

• The angle measured from the horizon or horizontal line,

• Angle of Elevation

• The angle measured from the horizon or horizontal line,

down

up

Examples

• Example 8

An inquisitive math student is standing 25 feet from the base of the Washington Monument. The angle of elevation from her horizontal

line of sight is 87.4°. If her “eye height” is 5ft, how tall is the monument?

87.4°25 5

x 𝒕𝒂𝒏 𝟖𝟕. 𝟒 =𝒙

𝟐𝟓

𝟐𝟓 ∗ 𝒕𝒂𝒏 𝟖𝟕. 𝟒 = 𝒙

𝟓𝟓𝟎. 𝟓𝟒 = 𝒙

𝟓𝟓𝟓. 𝟓𝟒 = 𝒚