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Trigonometry - Trigonometry of Triangles

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    Study Guide Trigonometry – Trigonometry of Triangles 1. Law of Cosines Figure 1 Reference triangle for Law of Cosines. In the previous section, you learned how to solve right triangles . Now, we’re going a step further. In this section (and the next one), you’ll learn formulas that work for any triangle — whether it’s right , acute , or obtuse . Let’s start by setting up the triangle. • Let α, β, and γ be the three angles of a triangle. • Let a, b, and c be the lengths of the sides opposite α, β, and γ , respectively. With this setup, we can write three very important formulas. 1.1 The Law of Cosines Formulas The Law of Cosines consists of these three equations: Each equation relates one side of the triangle to the other two sides and the angle between them . Together, these formulas allow us to solve triangles that are not right triangles.

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    Study Guide 1.2 Where the Law of Cosines Comes From These formulas come from the distance formula and can be understood using a coordinate - plane diagram. Imagine placing the triangle on a graph: • Point B is on the x - axis at ( a, 0 ) • Point C is at the origin ( 0, 0 ) • Point A is somewhere above the x - axis at ( x, y ) This setup makes it easier to use trigonometry and distance formulas. 1.3 Using Trigonometry to Find Coordinates From the diagram, we can use basic trigonometry: Solving for (x) and (y): So, the coordinates of point A are: 1.4 Applying the Distance Formula Now we find the distance from point A to point B using the distance formula: Substitute the expressions for (x) and (y): Expand and simplify:

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    Study Guide Since (sin 2 γ + cos 2 γ = 1), this becomes: This is exactly one of the Law of Cosines formulas. Important Notes • The Law of Cosines works even if angle γ is obtuse . • If ( γ = 90 °), then (cos 90 ° = 0). • The formula becomes: • This is the Pythagorean Theorem , which shows that the Law of Cosines is a generalization of it. • By placing a different point at the origin, you can derive the other two formulas in the same way. Why the Law of Cosines Matters One especially useful situation is when all three sides of a triangle are known . In that case, the Law of Cosines allows you to find the angles and completely solve the triangle. In short, the Law of Cosines is a powerful tool that lets you solve any triangle , not just right ones — and that makes it an essential part of trigonometry. Example 1: Finding an Angle Using the Law of Cosines Suppose α, β, and γ are the angles of a triangle, and a, b, and c are the lengths of the sides opposite those angles. You are given: • a = 12 • b = 7 • c = 6 Your goal is to find the measure of angle β .

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    Study Guide Step 1: Choose the Correct Law of Cosines Formula Since we are looking for angle β , we use the Law of Cosines formula that includes β: Step 2: Solve the Formula for cos β Rearrange the equation to isolate cos β: Now substitute the known values: Step 3: Find Angle β Since cos β > 0 and angles in a triangle are less than (180°), angle β must be acute less than (90°). Step 4: Find Angle α ( Using the Same Idea) Now let’s find angle α using a similar process. Start with the Law of Cosines formula that includes α: Solve for cos α :

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    Study Guide Substitute the given values: Step 5: Interpret the Result Because (cos α < 0), angle α must be greater than (90 ° ) (an obtuse angle). Step 6: Find the Final Angle γ The angles of a triangle always add up to (180°). So, Example 2: Finding a Side Length Using the Law of Cosines Figure 2 Drawing for Example 2. In this example, you are given a triangle where two sides and the included angle are known. Your goal is to find the length of side (b) .

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    Study Guide From the diagram (Figure 2): • One side has length (a = 11) • Another side has length (c = 10) • The angle between them is ( β = 71°) Step 1: Choose the Correct Law of Cosines Formula Since you are solving for side (b) and the known angle is ( β) , use this form of the Law of Cosines: Step 2: Substitute the Given Values Plug in the known side lengths and angle: Now simplify step by step: Step 3: Solve for (b) To find the length of side (b), take the square root: Final Answer

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    Study Guide Example 3: Finding the Area of a Triangle Figure 3 Drawing for Example 3. In this example, you are asked to find the area of the triangle from Example 2 . To make the calculations easier, first reposition the triangle so that the known angle is in standard position (as shown in Figure 3). This lets us use basic trigonometry to find the height. Step 1: Identify the Base From the diagram: • The base of the triangle lies along the x - axis. • The length of the base is: Step 2: Find the Height of the Triangle To find the area, we also need the height of the triangle. Look at the right triangle formed by dropping a vertical line from the top vertex. Using the sine of the given angle:

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    Study Guide Solve for (h): So, the height of the triangle is approximately 9.455 units . Step 3: Use the Area Formula Now apply the formula for the area of a triangle: Substitute the known values: Final Answer Summary • When you know a side and an angle, you can use trigonometry to find the height. • Repositioning the triangle can make calculations much simpler. • Once you have the base and height, finding the area is straightforward. • Sketch the triangle. • Use the Pythagorean theorem to find the hypotenuse. • Apply the definitions carefully. • Pay attention to signs based on the quadrant.

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    Study Guide 2. Law of Sines Figure 1 Reference triangles for Law of Sines. So far, you’ve learned how to use the Law of Cosines to solve triangles. It works well — but only in certain situations. In this section, we’ll learn a new tool called the Law of Sines , which gives us more flexibility when working with general triangles (triangles that are not right triangles). We’ll build the Law of Sines step by step, using a diagram to understand where it comes from and why it works. 2.1 Setting Up the Triangle Look at Figure 1 , which shows two triangles labeled (a) and (b). • The triangle has vertices A, B, and C • The sides opposite angles α, β, γ are labeled ( a, b, c ) • A line segment CD is drawn straight down from point C to side AB This line segment CD is an altitude , meaning it meets side AB at a right angle. Because of this: • Triangle ACD is a right triangle • Triangle BCD is also a right triangle This is important because it allows us to use basic trigonometry (like sine).

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    Study Guide 2.2 Using Sine in Triangle (a) In triangle ACD , focus on angle ( α ). From right - triangle trigonometry: Rewriting this: Now look at triangle BCD . The reference angle for β is the same, so: This gives: Since both expressions equal ( h ) , we can set them equal to each other: Divide both sides by ( ab ) : 2.3 Repeating the Process with Angle γ Now, draw an altitude from point A instead. Using the same idea: Divide both sides by ( ac ) :

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