April 21, 2017 The Law of Sines Topic List for Test Geometry April 21, 2017 The Law of Sines Topic List for Test
What is the length of the hypotenuse? How tall is the triangle? Do Now: Answer ALL FIVE What is the length of the hypotenuse? How tall is the triangle? How wide is it? tan 𝐴 = 𝑚∠𝐴= A
I can apply the Pythagorean Theorem and the trigonometric ratios Chapter Objective I can apply the Pythagorean Theorem and the trigonometric ratios
Today’s Objectives Apply the Law of Sines Review topics for the test
Here’s another triangle… Divide the sine by the opposite side: sin 40.4° /4.4 = sin 35° /3.9= sin 104.6° /6.5=
Here’s another triangle… Divide the sine by the opposite side: sin 54.2° /5.0 = sin 73.3° /5.9= sin 52.5° /4.9=
sin 𝐴 𝑎 = sin 𝐵 𝑏 = sin 𝐶 𝑐 The Law of Sines For ANY triangle, the ratio of the sine of each angle and its opposite side is constant: sin 𝐴 𝑎 = sin 𝐵 𝑏 = sin 𝐶 𝑐
Solve for BC sin 93° 15 = sin 48° 𝐵𝐶 𝐵𝐶= 15 sin 48° sin 93° 𝐵𝐶≈11.16 Example Solve for BC sin 93° 15 = sin 48° 𝐵𝐶 𝐵𝐶= 15 sin 48° sin 93° 𝐵𝐶≈11.16
Solve for 𝑚∠𝑆 sin 𝑆 11 = sin 120° 18 𝑚∠𝑆= sin −1 11 sin 120° 18 Example Solve for 𝑚∠𝑆 sin 𝑆 11 = sin 120° 18 𝑚∠𝑆= sin −1 11 sin 120° 18 𝑚∠𝑆≈31.95°
Solve for x
Solve for x
Solve for x
Practice
The Pythagorean Theorem A triangle is a right triangle IF AND ONLY IF it satisfies the equation: 𝑎2 + 𝑏2 = 𝑐2
Pythagorean Triples A Pythagorean triple is a set of positive integers that form a right triangle. Some common Pythagorean triples include: 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25
Review Special Triangles
SINE The ratio of the OPPOSITE LEG to the HYPOTENUSE INCREASES as the angle measure INCREASES sin 𝛼 = 𝑂 𝐻
COSINE The ratio of the ADJACENT LEG to the HYPOTENUSE DECREASES as the angle measure INCREASES cos 𝛼 = 𝐴 𝐻
TANGENT The ratio of the OPPOSITE LEG to the ADJACENT LEG INCREASES as the angle measure INCREASES tan 𝛼 = 𝑂 𝐴
Sin, Cos, Tan ANGLE MEASURE RATIO OF LENGTHS Sin-1, Cos-1, Tan-1
Think about… This is called the angle of ELEVATION. This is called the angle of DEPRESSION.