Area = Write the area formula. Area = Substitute c sin A for h. This formula allows you to determine the area of a triangle if you know the lengths of two of its sides and the measure of the angle between them.

Example 1: Determining the Area of a Triangle Find the area of the triangle. Round to the nearest tenth. Area = ab sin C Write the area formula. Substitute 3 for a, 5 for b, and 40° for C. Use a calculator to evaluate the expression. ≈ 4.820907073 The area of the triangle is about 4.8 m2.

Example 2A: Using the Law of Sines Solve the triangle. Round to the nearest tenth. Step 1. Find the third angle measure. mD + mE + mF = 180° Triangle Sum Theorem. Substitute 33° for mD and 28° for mF. 33° + mE + 28° = 180° mE = 119° Solve for mE.

Example 2A Continued Step 2 Find the unknown side lengths. sin D sin F d f = sin E sin F e f = Law of Sines. sin 33° sin 28° d 15 = sin 119° sin 28° e 15 = Substitute. Cross multiply. d sin 28° = 15 sin 33° e sin 28° = 15 sin 119° e = 15 sin 119° sin 28° e ≈ 27.9 d = 15 sin 33° sin 28° d ≈ 17.4 Solve for the unknown side.

Example 2B: Using the Law of Sines Q r Solve the triangle. Round to the nearest tenth. Step 2 Find the unknown side lengths. sin P sin Q p q = Law of Sines. sin P sin R p r = sin 105° sin 36° 10 q = sin 105° sin 39° 10 r = Substitute. q = 10 sin 36° sin 105° ≈ 6.1 r = 10 sin 39° ≈ 6.5

Check It Out! Example 2a Find the missing measures in the triangle. Round to the nearest tenth. Step 1 Find the third angle measure. mH + mJ + mK = 180° Substitute 42° for mH and 107° for mJ. 42° + 107° + mK = 180° mK = 31° Solve for mK.

Check It Out! Example 2a Continued Step 2 Find the unknown side lengths. sin H sin J h j = sin K sin H k h = Law of Sines. sin 42° sin 107° h 12 = sin 31° sin 42° k 8.4 = Substitute. Cross multiply. h sin 107° = 12 sin 42° 8.4 sin 31° = k sin 42° k = 8.4 sin 31° sin 42° k ≈ 6.5 h = 12 sin 42° sin 107° h ≈ 8.4 Solve for the unknown side.

Example 3 Given the measurements: a = 50, b = 20, and mA = 28°, find the other measures in the triangle. Round to the nearest tenth. Law of Sines Substitute. Solve for sin B.

Example 3 Continued So: Since m B = Sin-1 Step 3 Find the other unknown measures of the triangle. Solve for mC. 28° + 10.8° + mC = 180° mC = 141.2°

Example 3 Continued Now, Solve for c. Law of Sines Substitute. Solve for c. c ≈ 66.8

Inverse Sine Symbols/Notation: sin-1 or Arcsin Always used to find an angle given the ratio of sides oppostie/hypotenuse In order to view the inverse Sine as a function, we must review what we know about functions and their inverses

Inverse Sine Soooo….What interval does this? Q: How can we determine if a graph represents a function? Q: How can we determine if the graph of a function has an Inverse that is also a function? So: We restrict the domain… 1)Include 1st quadrant 2)Include the entire range of the function 3)Continuous interval Soooo….What interval does this?

Domain and Range Restricted Sine Function: Domain: -π/2 ≤ x ≤ π/2 Range: -1 ≤ y ≤ 1 Inverse Sine Function: Domain: -1 ≤ x ≤ 1 Range: -π/2 ≤ x ≤ π/2

Examples Evaluate the following: a) b) c)