Trig Blueprint Slides Trig Blueprint
Identity Review & Mastery
Reciprocal • Pythagorean • Sum & Difference
01: The Foundation
Reciprocal Identities
\(\csc \theta\) \( \frac{1}{\sin \theta} \)
\(\sec \theta\) \( \frac{1}{\cos \theta} \)
\(\cot \theta\) \( \frac{1}{\tan \theta} \)
Quotient Identities
\(\tan \theta\) \( \frac{\sin \theta}{\cos \theta} \)
\(\cot \theta\) \( \frac{\cos \theta}{\sin \theta} \)
02: The Structure
\(\sin^2 \theta + \cos^2 \theta = 1\)
The main structural beam. Divide by \(\sin^2 \theta\) or \(\cos^2 \theta\) to derive the remaining identities.
\(1 + \tan^2 \theta = \sec^2 \theta\)
\(1 + \cot^2 \theta = \csc^2 \theta\)
03: Sum & Difference
Sine Formula
\(\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B\)
Cosine Formula
\(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\)
Tangent Formula
\( \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \)
Mastery Strategy
1
Simplify by converting to Sine & Cosine.
2
Match Squared Terms with Pythagorean identities.
3
Use algebra: Factoring and common denominators.
4
Apply Sum/Diff to expand or find exact values.
Trig Identity Quiz IDENTITY BLUEPRINT QUIZ
Trigonometry Mastery Assessment
NAME:
DATE:
Part I: Multiple Choice. Choose the best simplified form for each expression.
1. Simplify: \(\cos x \tan x\)
A) \(\cot x\)
B) \(\sin x\)
C) \(\sec x\)
D) \(1\)
2. Value of \(\csc^2 \theta - \cot^2 \theta\):
A) \(-1\)
B) \(\tan^2 \theta\)
C) \(1\)
D) \(\sin^2 \theta\)
3. Simplify: \((1 - \cos^2 x)(\csc x)\)
A) \(\sin x\)
B) \(\cos x\)
C) \(\tan x\)
D) \(1\)
4. Exact value of \(\sin(15^\circ)\):
A) \(\frac{\sqrt{6} + \sqrt{2}}{4}\)
B) \(\frac{\sqrt{2} - \sqrt{6}}{4}\)
C) \(\frac{\sqrt{6} - \sqrt{2}}{4}\)
D) \(\frac{\sqrt{2} + \sqrt{6}}{4}\)
5. Simplify: \(\frac{\sec \theta}{\csc \theta}\)
A) \(\tan \theta\)
B) \(\cot \theta\)
C) \(1\)
D) \(\sin \theta \cos \theta\)
6. \(\cos(A - B)\) expansion is:
A) \(\cos A \cos B - \sin A \sin B\)
B) \(\cos A \cos B + \sin A \sin B\)
C) \(\sin A \cos B - \cos A \sin B\)
D) \(\sin A \sin B - \cos A \cos B\)
7. Simplify: \(\tan^2 x - \sec^2 x\)
A) \(1\)
B) \(-1\)
C) \(\tan^2 x\)
D) \(0\)
8. \(\cos(75^\circ)\) is equal to:
A) \(\sin 15^\circ\)
B) \(\cos(45^\circ + 30^\circ)\)
C) \(\sin 75^\circ\)
D) Both A and B
9. Simplify: \(\sin(-x)\cot(-x)\)
A) \(\cos x\)
B) \(-\cos x\)
C) \(\sin x\)
D) \(-\sin x\)
10. Expand \(\tan(x + \frac{\pi}{4})\):
A) \(\frac{\tan x + 1}{1 - \tan x}\)
B) \(\frac{\tan x - 1}{1 + \tan x}\)
C) \(\frac{1 - \tan x}{1 + \tan x}\)
D) \(\tan x + 1\)
Part II: Multi-Step Proofs. Show every step logically.
1. Prove: \(\cos x (\sec x - \cos x) = \sin^2 x\)
2. Prove: \(\frac{\sin \theta}{1 + \cos \theta} + \frac{1 + \cos \theta}{\sin \theta} = 2\csc \theta\)
3. Prove: \(\cos(x + y) + \cos(x - y) = 2\cos x \cos y\)
4. Prove: \(\sin(x + \pi) = -\sin x\)
5. Prove: \(\frac{\tan x + \cot x}{\sec x \csc x} = 1\)
6. Prove: \(\sin(A+B)\sin(A-B) = \sin^2 A - \sin^2 B\)
7. Prove: \(\frac{1}{\sec x + 1} + \frac{1}{\sec x - 1} = 2\csc x \cot x\)
8. Prove: \(\tan x + \cot x = \sec x \csc x\)
Identity Shuffle Practice Identity Shuffle Practice
NAME: ___________________________
Part I: Identity Shuffle (MC)
1. Simplify \(\sec \theta \cot \theta\):
A) \(\csc \theta\)
B) \(\sin \theta\)
C) \(\cos \theta\)
D) \(1\)
2. Value of \(\sin^2 x + \cos^2 x + \tan^2 x\):
A) \(1\)
B) \(\sec^2 x\)
C) \(\csc^2 x\)
D) \(\cot^2 x\)
3. \(\cos(x + \frac{\pi}{2})\) is:
A) \(\sin x\)
B) \(-\sin x\)
C) \(\cos x\)
D) \(-\cos x\)
4. Simplify \((1 - \cos^2 \theta)\csc^2 \theta\):
A) \(\sin \theta\)
B) \(1\)
C) \(\cos \theta\)
D) \(\tan \theta\)
5. \(\sin(A)\cos(B) + \cos(A)\sin(B) =\)
A) \(\cos(A+B)\)
B) \(\sin(A+B)\)
C) \(\sin(A-B)\)
D) \(\cos(A-B)\)
6. \(\cos 15^\circ\) find exactly:
Show work area
7. \(\sin 75^\circ\) find exactly:
Show work area
8. \(\tan 105^\circ\) find exactly:
Show work area
9. \(\cot x \sin x\) simplify:
A) \(\cos x\)
B) \(\sin x\)
C) \(\tan x\)
D) \(1\)
10. \(\cos^2 \theta(1 + \tan^2 \theta) =\)
A) \(\sin^2 \theta\)
B) \(1\)
C) \(\cos^2 \theta\)
D) \(\sec^2 \theta\)
Part II: Practice Proofs (1-4)
1. \(\sin x \cot x = \cos x\)
2. \((\sec x - 1)(\sec x + 1) = \tan^2 x\)
3. \(\frac{1}{\tan x} + \tan x = \sec x \csc x\)
4. \(\sin(x - y)\cos y + \cos(x - y)\sin y = \sin x\)
Part II: Practice Proofs (5-8)
5. \(\frac{\cos^2 x - \sin^2 x}{1 - \tan^2 x} = \cos^2 x\)
6. \(\sin(A-B) + \sin(A+B) = 2\sin A \cos B\)
7. \(\frac{1 + \sin x}{\cos x} + \frac{\cos x}{1 + \sin x} = 2\sec x\)
8. \(\cos^4 x - \sin^4 x = \cos^2 x - \sin^2 x\)
Practice Answer Key
1: A
2: B
3: B
4: B
5: B
6: \(\frac{\sqrt{6}+\sqrt{2}}{4}\)
7: \(\frac{\sqrt{6}+\sqrt{2}}{4}\)
8: \(-2-\sqrt{3}\)
9: A
10: B
Trig Mastery Keys Teacher Solution Blueprint
Mastery Key: Assessment & Practice
Assessment MCQ Key
1. B (\(\sin x\))
2. C (\(1\))
3. A (\(\sin x\))
4. C (\(\frac{\sqrt{6}-\sqrt{2}}{4}\))
5. A (\(\tan \theta\))
6. B (\(\cos A \cos B + \sin A \sin B\))
7. B (\(-1\))
8. D (Both A and B)
9. A (\(\cos x\))
10. A (\(\frac{\tan x + 1}{1 - \tan x}\))
Assessment Proof Solutions
1. \(\cos x (\frac{1}{\cos x} - \cos x) = 1 - \cos^2 x = \sin^2 x\)
2. \(\frac{\sin^2 \theta + (1 + \cos \theta)^2}{\sin \theta(1 + \cos \theta)} = \frac{\sin^2 \theta + 1 + 2\cos \theta + \cos^2 \theta}{\sin \theta(1 + \cos \theta)} = \frac{2 + 2\cos \theta}{\sin \theta(1 + \cos \theta)} = \frac{2(1 + \cos \theta)}{\sin \theta(1 + \cos \theta)} = 2\csc \theta\)
3. \((\cos x \cos y - \sin x \sin y) + (\cos x \cos y + \sin x \sin y) = 2\cos x \cos y\)
4. \(\sin x \cos \pi + \cos x \sin \pi = \sin x(-1) + \cos x(0) = -\sin x\)
5. \(\frac{(\sin x / \cos x + \cos x / \sin x)}{\sec x \csc x} = \frac{(\sin^2 x + \cos^2 x) / (\sin x \cos x)}{1 / (\sin x \cos x)} = \frac{1/\sin x \cos x}{1/\sin x \cos x} = 1\)
6. \((\sin A \cos B + \cos A \sin B)(\sin A \cos B - \cos A \sin B) = \sin^2 A \cos^2 B - \cos^2 A \sin^2 B\). Substitute \(\cos^2 = 1-\sin^2\) to get \(\sin^2 A - \sin^2 B\).
7. \(\frac{\sec x - 1 + \sec x + 1}{\sec^2 x - 1} = \frac{2\sec x}{\tan^2 x} = \frac{2/\cos x}{\sin^2 x / \cos^2 x} = \frac{2\cos x}{\sin^2 x} = 2\csc x \cot x\)
8. \(\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x} = \csc x \sec x\)
Practice Solution Blueprint
Practice Proof #3: \(\cot x + \tan x = \frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = \frac{\cos^2 x + \sin^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x} = \csc x \sec x\)
Practice Proof #4: Expansion of \(\sin((x-y) + y) = \sin x\).
Practice Proof #5: \(\frac{\cos^2 x - \sin^2 x}{1 - \tan^2 x} = \frac{\cos^2 x - \sin^2 x}{(\cos^2 x - \sin^2 x) / \cos^2 x} = \cos^2 x\).
