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Trigonometry - Trigonometric Identities

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    Study Guide Trigonometry – Trigonometric Identities 1. Addition Identities In the previous section, we worked with basic identities that involved only one variable . Now we’re going a step further. In this section, we introduce identities that involve two angles . These are called trigonometric addition identities . Here they are: These four formulas are sometimes called: • Sum identity for sine • Difference identity for sine • Sum identity for cosine • Difference identity for cosine They may look a little heavy at first, but don’t worry — once you see them in action, they start to feel very natural. These identities can be proven using basic trigonometric facts and the distance formula in the coordinate plane. In this section, we’ll focus mainly on understanding how to use them. Example 1 : Rewrite as a trigonometric function involving only one angle.

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    Study Guide Figure 1 Drawing for Example 1. Step 1: Recognize the Pattern Look closely at the expression: This matches the sum identity for sine : So we can rewrite the expression as: Step 2: Simplify the Angle Now think about where (210 ° ) lies. It’s in the third quadrant , where sine values are negative . So:

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    Study Guide And since: We get: Final Answer: Nice and clean once you spot the identity! 1 .1 Reduction Formulas for Cosine The next few examples show how addition identities help us rewrite trigonometric functions of large angles (like 180° or 360°) in simpler forms. These are called reduction formulas . Example 2 : Verify that: Step 1: Use the Cosine Difference Identity Let ( α = 180 ° ) and ( β = x). Step 2: Substitute Known Values

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    Study Guide So: Example 3 : Verify that: Step 1: Use the Cosine Sum Identity Substitute ( α = 180 ° ), ( β = x): Step 2: Substitute Known Values Example 4 : Verify that: Using the cosine difference identity: Now use: [ So:

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    Study Guide Why These Matter These three examples give us reduction formulas for cosine . They help us rewrite cosines of large angles (greater than 90°) as simpler expressions involving acute angles. This makes calculations much easier. 1.2 Reduction Formulas for Sine Let’s do the same thing for sine. Example 5 : Verify that: Step 1: Use the Sine Difference Identity Substitute ( α = 180 ° ), ( β = x): Step 2: Substitute Known Values Example 6 : Verify that: Using the sine sum identity:

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    Study Guide Substitute values: Example 7 : Verifying a Reduction Formula We want to verify: Start with the sine subtraction formula: Now apply it: We know: • (sin 360° = 0) • (cos 360° = 1) So, What Are Reduction Formulas? Reduction formulas rewrite trig functions of large angles using acute angles (angles less than (90°). They work for both degrees and radians and make trig problems much easier to solve.

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    Study Guide 1.3 Common Reduction Formulas For Cosine Or in radians: For Sine Or in radians: These help you rewrite trig functions using smaller, easier angles. Example 8 : Double - Angle Identity Start with:

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    Study Guide Use the sine addition formula: So, Example 9 : Writing as One Variable Rewrite: Use the cosine identity: Here, Example 10 : Rewrite Using Sine Write cos 303° as sin β, where 0° < β < 90°. Now use: So, (cos 303° = sin 33°).

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    Study Guide Example 11 : Rewrite Using Cosine Write (sin 234°) in the form (cos β ) , where 0° < β < 90°. Now use: So, (sin 234° = - cos 36°). Example 12 : Finding sin α + β We are given: Both angles are in the fourth quadrant , where: • sine is negative • cosine is positive Step 1: Find missing values Using (sin 2 x + cos 2 x = 1 ) : For ( α ) :

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    Study Guide For ( β ) : Since sine is negative in quadrant IV: Step 2: Use the sine addition formula Substitute:

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