Chapter 7-5 Notes
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