Unlocking Trigonometry: Mastering the Six Trig Functions

Algebra 2 Grades High School 4:48 Video
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Lesson Description

A comprehensive lesson on evaluating the six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) using right triangles and the SOH CAH TOA mnemonic. Includes practice problems and a review of the Pythagorean theorem.

Video Resource

Evaluate the Six Trigonometric Functions of the Angle

Mario's Math Tutoring

Duration: 4:48
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Key Concepts

  • Trigonometric Ratios (Sine, Cosine, Tangent)
  • Reciprocal Trigonometric Ratios (Cosecant, Secant, Cotangent)
  • SOH CAH TOA mnemonic
  • Pythagorean Theorem
  • Right Triangle Trigonometry

Learning Objectives

  • Students will be able to identify the opposite, adjacent, and hypotenuse sides of a right triangle with respect to a given angle.
  • Students will be able to calculate the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) of an angle in a right triangle.
  • Students will be able to use the Pythagorean theorem to find a missing side length of a right triangle.
  • Students will be able to apply the SOH CAH TOA mnemonic to determine trigonometric ratios.

Educator Instructions

  • Introduction to Trigonometric Ratios (5 mins)
    Begin by defining trigonometry as the study of the ratios of sides in a triangle. Introduce the sine, cosine, and tangent ratios and the SOH CAH TOA mnemonic. Explain what each part of SOH CAH TOA means (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent).
  • Reciprocal Trigonometric Ratios (5 mins)
    Define the reciprocal trigonometric ratios: cosecant, secant, and cotangent. Explain that cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent. Show that cosecant = Hypotenuse/Opposite, secant = Hypotenuse/Adjacent, and cotangent = Adjacent/Opposite.
  • Introductory Example (10 mins)
    Work through a simple example using a 3-4-5 right triangle. Given an angle theta, identify the opposite, adjacent, and hypotenuse sides. Calculate all six trigonometric functions for that angle. Emphasize the importance of positioning oneself at the correct angle when identifying the sides.
  • Pythagorean Theorem Review (5 mins)
    Briefly review the Pythagorean theorem (a² + b² = c²) and its application in finding the missing side of a right triangle. Explain that the hypotenuse is always 'c' and is opposite the right angle.
  • Example Problems (15 mins)
    Work through multiple example problems with varying side lengths. In each problem: 1) Identify the missing side using the Pythagorean theorem. 2) Identify the opposite, adjacent, and hypotenuse sides with respect to the given angle. 3) Calculate all six trigonometric functions. 4) Rationalize denominators when necessary.
  • Mistakes to Avoid (5 mins)
    Reinforce that the Hypotenuse is always opposite the right angle, and demonstrate how it is a common error to incorrectly label the adjacent and hypotenuse sides.

Interactive Exercises

  • Triangle Side Identification
    Present students with various right triangles with a marked angle. Ask them to identify the opposite, adjacent, and hypotenuse sides for each triangle.
  • Trig Function Calculation
    Provide students with right triangles with given side lengths and a marked angle. Have them calculate all six trigonometric functions for that angle.
  • Missing Side Calculation
    Provide students with a right triangle with two known side lengths and ask them to calculate the missing side length using the Pythagorean theorem.

Discussion Questions

  • How does the SOH CAH TOA mnemonic help you remember the trigonometric ratios?
  • What is the relationship between sine and cosecant, cosine and secant, and tangent and cotangent?
  • How does knowing two sides of a right triangle allow you to find all six trigonometric functions for a given angle?
  • Why is it important to correctly identify the opposite, adjacent, and hypotenuse sides before calculating the trigonometric ratios?

Skills Developed

  • Application of Trigonometric Ratios
  • Problem-Solving using Pythagorean Theorem
  • Analytical Thinking
  • Memorization and Recall (SOH CAH TOA)
  • Attention to Detail

Multiple Choice Questions

Question 1:

What does SOH stand for in the SOH CAH TOA mnemonic?

Correct Answer: Sine = Opposite/Hypotenuse

Question 2:

Which trigonometric function is the reciprocal of cosine?

Correct Answer: Secant

Question 3:

If the opposite side of an angle in a right triangle is 5 and the hypotenuse is 13, what is the sine of the angle?

Correct Answer: 5/13

Question 4:

Which side is adjacent to the right angle?

Correct Answer: Hypotenuse

Question 5:

In a right triangle, if the adjacent side to an angle is 8 and the opposite side is 6, what is the tangent of the angle?

Correct Answer: 6/8

Question 6:

What is the cosecant of angle theta if sin(theta) = 3/5?

Correct Answer: 5/3

Question 7:

If cos(theta) = 12/13, what is the secant of angle theta?

Correct Answer: 13/12

Question 8:

The acronym SOH CAH TOA helps us remember:

Correct Answer: The ratios of trig functions

Question 9:

If tan(theta) = 7/24, what is the cotangent of angle theta?

Correct Answer: 24/7

Question 10:

Given a right triangle with legs of length 3 and 4, what is the length of the hypotenuse?

Correct Answer: 5

Fill in the Blank Questions

Question 1:

The sine of an angle is equal to the _________ side divided by the hypotenuse.

Correct Answer: opposite

Question 2:

The acronym _________ is used to remember the trigonometric ratios.

Correct Answer: SOH CAH TOA

Question 3:

The _________ theorem can be used to find the missing side length of a right triangle.

Correct Answer: Pythagorean

Question 4:

The cosecant is the reciprocal of the _________ function.

Correct Answer: sine

Question 5:

Tangent is the ratio of the _______ side to the adjacent side.

Correct Answer: opposite

Question 6:

Adjacent means _______ to.

Correct Answer: next

Question 7:

The secant function is the reciprocal of the _________ function.

Correct Answer: cosine

Question 8:

The cotangent function is the reciprocal of the _______ function.

Correct Answer: tangent

Question 9:

In a right triangle, the side opposite the right angle is called the _______.

Correct Answer: hypotenuse

Question 10:

Cosine is the ratio of the adjacent side to the _________.

Correct Answer: hypotenuse

Educational Standards

A2.F.1.1:Use algebraic, interval, and set notations to specify the domain and range of various types of functions, and evaluate a function at a given point in its domain. A2.A.2.5:Rewrite algebraic expressions involving radicals and rational exponents using the properties of exponents.

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