Mathematics
Grade 11
15 min
Trigonometric identities: Set 1
Trigonometric identities: Set 1
What you'll learn
- Identify the nearest dollar when given a money amount less than $10 containing cents.
- Explain the rule for rounding money amounts to the nearest dollar (50 cents or more rounds up, less than 50 cents rounds down).
- Apply rounding to estimate the total cost of two items, each priced under $5, to the nearest dollar.
- Solve word problems involving rounding money to the nearest dollar, and explain whether the rounded amount is an overestimate or underestimate.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Recall and apply the reciprocal and quotient identities.
State and use the three Pythagorean identities in their various forms.
Simplify complex trigonometric expressions using fundamental identities.
Prove basic trigonometric identities by manipulating one side of the equation to match the other.
Express one trigonometric function in terms of another.
Factor trigonometric expressions and combine trigonometric fractions.
Ever wondered how video game engines create realistic lighting and shadows? 🎮 It involves complex calculations built on the foundational 'rules' of trigonometry we're about to explore!
Trigonometric identities are equations that are true for all values of the variables involved. They are the fundamental rules of trigonome...
2
Key Concepts & Vocabulary
TermDefinitionExample
Trigonometric IdentityAn equation involving trigonometric functions that is true for all values of the variable for which both sides of the equation are defined.The equation `sin²(θ) + cos²(θ) = 1` is an identity because it is true for any angle θ.
Reciprocal IdentitiesIdentities that define the relationship between a trigonometric function and its multiplicative inverse.`csc(θ) = 1/sin(θ)`. If `sin(θ) = 1/2`, then `csc(θ) = 2`.
Quotient IdentitiesIdentities that express one trigonometric function as a ratio of two others.`tan(θ) = sin(θ)/cos(θ)`. This shows the relationship between tangent, sine, and cosine.
Pythagorean IdentitiesA set of three fundamental identities derived from applying the Pythagorean theorem to a right triangle in the unit circle.`1 + tan²(θ) =...
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Core Formulas
Reciprocal & Quotient Identities
`csc(θ) = \frac{1}{sin(θ)}` | `sec(θ) = \frac{1}{cos(θ)}` | `cot(θ) = \frac{1}{tan(θ)}`
`tan(θ) = \frac{sin(θ)}{cos(θ)}` | `cot(θ) = \frac{cos(θ)}{sin(θ)}`
Use these identities as your first step to simplify expressions, typically by converting all functions into terms of sine and cosine.
The Pythagorean Identities
1. `sin²(θ) + cos²(θ) = 1`
2. `1 + tan²(θ) = sec²(θ)`
3. `1 + cot²(θ) = csc²(θ)`
These are crucial for substitutions, especially when you see squared trigonometric functions. Remember they can be rearranged, for example: `sin²(θ) = 1 - cos²(θ)`.
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Challenging
Simplify the expression (sec²(x) - 1)(cot²(x)).
A.tan²(x)
B.sin²(x)
C.cot²(x)
D.1
Challenging
Simplify the expression (sin⁴(θ) - cos⁴(θ)) / (sin²(θ) - cos²(θ)).
A.1
B.sin(θ) - cos(θ)
C.sin(θ) + cos(θ)
D.tan²(θ) - 1
Challenging
Simplify the expression (1 / (sec(x) + tan(x))) + (1 / (sec(x) - tan(x))).
A.2cos(x)
B.2tan(x)
C.2sec(x)
D.2
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Frequently asked questions
What grade level is "Trigonometric identities: Set 1"?
Trigonometric identities: Set 1 is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Trigonometric identities: Set 1?
You'll be able to: Identify the nearest dollar when given a money amount less than $10 containing cents; Explain the rule for rounding money amounts to the nearest dollar (50 cents or more rounds up, less than 50 cents rounds down); Apply rounding….
Is "Trigonometric identities: Set 1" free to practice?
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How many practice questions are included with Trigonometric identities: Set 1?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.