PRE MI UM ACT I VE- LE A RN IN G WO RK BOO K
Trigonometric Identities and Formulas
Angle Sum/Difference Formulas & Simplification Strategies
Trigonometry • Active-Learning Edition
How to use this workbook
1. Read the concept explanation. 2. Study the worked example. 3. Try the practice problems yourself.
4. Check your work against the Answer Key. 5. Revisit anything you missed.
, 1. Angle Sum and Difference Formulas
Sine Formulas
Term Definition
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
sin(x − y) = sin(x)cos(y) − cos(x)sin(y)
Cosine Formulas
Term Definition
cos(x + y) = cos(x)cos(y) − sin(x)sin(y)
cos(x − y) = cos(x)cos(y) + sin(x)sin(y)
Tangent Formulas
Term Definition
tan(x + y) = (tan(x) + tan(y)) / (1 − tan(x)tan(y))
tan(x − y) = (tan(x) − tan(y)) / (1 + tan(x)tan(y))
WORKED EXAMPLE — Calculating sin(75°)
Break 75° into known angles: 75° = 45° + 30°.
Apply the sine sum formula: sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°).
Plug in known values: = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4.
Final answer: sin(75°) = (√6 + √2) / 4.
WORKED EXAMPLE — Calculating cos(255°)
Break 255° into known angles: 255° = 180° + 75°.
Apply the identity cos(180° + θ) = −cos(θ): cos(255°) = −cos(75°).
This shows how angle transformations let you reduce any angle to a reference angle you already know.
✏ TRY IT YOURSELF
1. Calculate cos(75°) using the cosine sum formula (break into 45° + 30°).
2. Calculate sin(15°) using the sine difference formula (break into 45° − 30°).
3. Calculate tan(15°) using the tangent difference formula.
2. Fundamental Trigonometric Identities
Reciprocal Identities
Term Definition
csc(x) = 1 / sin(x)
sec(x) = 1 / cos(x)
cot(x) = 1 / tan(x)
Trigonometric Identities and Formulas
Angle Sum/Difference Formulas & Simplification Strategies
Trigonometry • Active-Learning Edition
How to use this workbook
1. Read the concept explanation. 2. Study the worked example. 3. Try the practice problems yourself.
4. Check your work against the Answer Key. 5. Revisit anything you missed.
, 1. Angle Sum and Difference Formulas
Sine Formulas
Term Definition
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
sin(x − y) = sin(x)cos(y) − cos(x)sin(y)
Cosine Formulas
Term Definition
cos(x + y) = cos(x)cos(y) − sin(x)sin(y)
cos(x − y) = cos(x)cos(y) + sin(x)sin(y)
Tangent Formulas
Term Definition
tan(x + y) = (tan(x) + tan(y)) / (1 − tan(x)tan(y))
tan(x − y) = (tan(x) − tan(y)) / (1 + tan(x)tan(y))
WORKED EXAMPLE — Calculating sin(75°)
Break 75° into known angles: 75° = 45° + 30°.
Apply the sine sum formula: sin(75°) = sin(45°)cos(30°) + cos(45°)sin(30°).
Plug in known values: = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4.
Final answer: sin(75°) = (√6 + √2) / 4.
WORKED EXAMPLE — Calculating cos(255°)
Break 255° into known angles: 255° = 180° + 75°.
Apply the identity cos(180° + θ) = −cos(θ): cos(255°) = −cos(75°).
This shows how angle transformations let you reduce any angle to a reference angle you already know.
✏ TRY IT YOURSELF
1. Calculate cos(75°) using the cosine sum formula (break into 45° + 30°).
2. Calculate sin(15°) using the sine difference formula (break into 45° − 30°).
3. Calculate tan(15°) using the tangent difference formula.
2. Fundamental Trigonometric Identities
Reciprocal Identities
Term Definition
csc(x) = 1 / sin(x)
sec(x) = 1 / cos(x)
cot(x) = 1 / tan(x)
