Unit 3: Trigθnθmetric Functions Lesson 2 – Evaluating Trig. Ratios

Thursday, May 4, 2017

Agenda

1. Evaluating Trigonometric Ratios For Any Angle i. Acute angles (last class)

ii. Obtuse Angles –greater than 90° BUT less than 180°

iii. Reflex Angles – Angles Greater than 180° BUT less than 360°

Recall: Primary Trigonometric Ratios

Given a Right Triangle with side lengths 𝑥, 𝑦, 𝑟,

and angle 𝜃, the primary trig ratios are:

sin 𝜃 =𝑦

𝑟

cos 𝜃 =𝑥

𝑟

tan 𝜃 =𝑦

𝑥

𝑥

𝑦𝑟

𝜃

Definition:

Standard Position: _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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Definition:

Related Acute Angle

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Principal Angle

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In this example, 𝛽 represents the related acute angle for 𝜃.

Why is the related acute angle important?

• The related acute angle is the shortest angular distance to thex-axis.

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• This is helpful when graphing or evaluating EXACT trigonometric ratios for angles greater than 90°

If the terminal arm of an angle in standard position lies in quadrants 2, 3, or 4, there exists a

related acute angle and a principal angle.

We can use these related acute angles to evaluate trig ratios of larger angles!

How do you calculate the related acute angle?• If the terminal arm is in quadrant 2,

𝑅𝑒𝑙𝑎𝑡𝑒𝑑 𝐴𝑐𝑢𝑡𝑒 𝐴𝑛𝑔𝑙𝑒 = 180°− 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒 𝐴𝑛𝑔𝑙𝑒

• If the terminal arm is in quadrant 3, 𝑅𝑒𝑙𝑎𝑡𝑒𝑑 𝐴𝑐𝑢𝑡𝑒 𝐴𝑛𝑔𝑙𝑒 = 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒 𝐴𝑛𝑔𝑙𝑒 − 180°

• If the terminal arm is in quadrant 4, 𝑅𝑒𝑙𝑎𝑡𝑒𝑑 𝐴𝑐𝑢𝑡𝑒 𝐴𝑛𝑔𝑙𝑒 = 360°− 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒 𝐴𝑛𝑔𝑙𝑒

Example 1

Determine the principal angle and the related acute angle for 𝜃 = 225°.

Example 2: Use the point P(0,1) to determine the values of sine, cosine, and tangent for 𝜃 = 90°.

Example 3: Determine the values of 𝜃 if

𝑐𝑠𝑐𝜃 = −2 3

3and 0° ≤ 𝜃 ≤ 360°

Lesson 3: Trigonometric Identities

Trigonometric Identities

• trigonometric identities are equalities that involve trigonometricratios and are true for every single value of the occurring variables where both sides of the equality are defined

Trigonometric Identities

How to work with Trig. Identities

To prove that a given trigonometric equation is an identity, both sides of the equation need to be shown to be equivalent. This can be done by

• simplifying the more complicated side until it is identical to the other side or manipulating both sides to get the same expression

• rewriting all trigonometric ratios in terms of x, y, and r

• rewriting all expressions involving tangent and the reciprocal trigonometric ratios in terms of sine and cosine

• applying the Pythagorean identity where appropriate

• using a common denominator or factoring as required

Homework:

• Read and make notes – pg 298-299 **CAST RULE ESPECIALLY**

• Pg. 292 #1 – 4

• Pg. 299 – 300 # 1 – 4, 5 (cd), 6, 10

• Pg. 304 #8 – 13

• Pg. 310 #1-8