UPCAT Mathematics — TrigonometryExam Answer Templates

Examiner tip

Always use exact fractional values for special angles unless specifically asked for decimal approximation

Model answer

sin 30° = 1/2

Answer structure

• Line 1: Direct value from special angle table [1 mark]

Scoring breakdown

Common mark deductions

• Writing decimal 0.5 instead of fraction 1/2
• Incorrect value

Key phrases to include

• 1/2
• exact value

Examiner tip

Always write the conversion formula first to show understanding of the method

Model answer

Given: Angle = 45°
To convert to radians:
Using the formula: radians = degrees × π/180°
= 45° × π/180°
= π/4 radians
Therefore, 45° = π/4 radians

Answer structure

• Line 1: State the conversion formula [1 mark]
• Line 2: Substitute and calculate correctly [1 mark]

Scoring breakdown

Common mark deductions

• Not showing the conversion formula
• Incorrect calculation
• Missing π in the answer

Key phrases to include

• π/180°
• π/4 radians
• conversion formula

Examiner tip

Always find the missing side first using Pythagorean theorem, then apply SOH-CAH-TOA systematically

Model answer

Given: Opposite side = 3 units, Hypotenuse = 5 units
To find: sin θ, cos θ, tan θ

First, find adjacent side using Pythagorean theorem:
adjacent² + opposite² = hypotenuse²
adjacent² + 3² = 5²
adjacent² = 25 - 9 = 16
adjacent = 4 units

Now:
sin θ = opposite/hypotenuse = 3/5
cos θ = adjacent/hypotenuse = 4/5
tan θ = opposite/adjacent = 3/4

Therefore: sin θ = 3/5, cos θ = 4/5, tan θ = 3/4

Answer structure

• Line 1-3: Find missing side using Pythagorean theorem [1 mark]
• Line 4: Calculate sin θ correctly [1 mark]
• Line 5-6: Calculate cos θ and tan θ correctly [1 mark]

Scoring breakdown

Common mark deductions

• Not finding the adjacent side first
• Incorrect trigonometric ratios
• Arithmetic errors

Key phrases to include

• Pythagorean theorem
• opposite/hypotenuse
• adjacent/hypotenuse
• opposite/adjacent

Examiner tip

Always consider the quadrant to determine the correct sign of trigonometric functions

Model answer

Given: sin θ = 3/5, θ is in first quadrant
To find: cos θ and tan θ

Using the identity: sin²θ + cos²θ = 1
(3/5)² + cos²θ = 1
9/25 + cos²θ = 1
cos²θ = 1 - 9/25 = 16/25
cos θ = ±4/5

Since θ is in first quadrant, cos θ > 0
Therefore, cos θ = 4/5

tan θ = sin θ/cos θ = (3/5)/(4/5) = 3/4

Therefore: cos θ = 4/5, tan θ = 3/4

Answer structure

• Line 1-4: Use Pythagorean identity to find cos²θ [1 mark]
• Line 5-6: Determine correct sign based on quadrant [1 mark]
• Line 7: Calculate tan θ using ratio identity [1 mark]

Scoring breakdown

Common mark deductions

• Not considering quadrant signs
• Arithmetic errors
• Not using correct trigonometric identity

Key phrases to include

• Pythagorean identity
• first quadrant
• sin²θ + cos²θ = 1
• tan θ = sin θ/cos θ

Examiner tip

Always start proofs with a fundamental identity and show each algebraic step clearly

Model answer

To prove: tan²θ + 1 = sec²θ

Starting with the fundamental identity: sin²θ + cos²θ = 1

Dividing both sides by cos²θ:
(sin²θ + cos²θ)/cos²θ = 1/cos²θ
sin²θ/cos²θ + cos²θ/cos²θ = 1/cos²θ
(sin θ/cos θ)² + 1 = (1/cos θ)²
tan²θ + 1 = sec²θ

Hence proved.

Answer structure

• Line 1: Start with fundamental Pythagorean identity [1 mark]
• Line 2-4: Divide by cos²θ and simplify correctly [1 mark]
• Line 5: Recognize tan θ and sec θ definitions [1 mark]

Scoring breakdown

Common mark deductions

• Starting with wrong identity
• Algebraic errors
• Not showing clear logical steps

Key phrases to include

• fundamental identity
• dividing by cos²θ
• tan θ = sin θ/cos θ
• sec θ = 1/cos θ
• Hence proved

Examiner tip

Always draw a clear diagram for word problems and use exact values for special angles

Model answer

Given: Length of ladder = 10 m, Angle with ground = 60°
To find: Height at which ladder touches the wall

From the right triangle formed:

• Hypotenuse = ladder length = 10 m
• Angle between ladder and ground = 60°
• Height = opposite side to the 60° angle

Using sin 60° = opposite/hypotenuse
sin 60° = height/10
√3/2 = height/10
height = 10 × √3/2 = 5√3 m

Therefore, the ladder touches the wall at a height of 5√3 meters.

Answer structure

• Line 1-2: Identify the right triangle and given values [1 mark]
• Line 3-4: Choose correct trigonometric ratio (sin 60°) [1 mark]
• Line 5-6: Calculate correctly using exact values [1 mark]

Scoring breakdown

Common mark deductions

• Using decimal values instead of exact values
• Wrong trigonometric ratio
• Not drawing or visualizing the triangle correctly

Key phrases to include

• right triangle
• sin 60° = √3/2
• opposite side
• 5√3 meters

Examiner tip

Remember the reference angle formulas for each quadrant: Q2: 180°-θ, Q3: θ-180°, Q4: 360°-θ

Model answer

Given: Angle = 150°
To find: Reference angle

150° lies in the second quadrant (90° < 150° < 180°)
For angles in the second quadrant, reference angle = 180° - angle
Reference angle = 180° - 150° = 30°

Therefore, the reference angle for 150° is 30°.

Answer structure

• Line 1: Identify which quadrant the angle is in [1 mark]
• Line 2: Apply correct formula and calculate [1 mark]

Scoring breakdown

Common mark deductions

• Wrong quadrant identification
• Using wrong formula for reference angle

Key phrases to include

• second quadrant
• 180° - angle
• reference angle = 30°

Examiner tip

Remember: in quadrant II, only sine and cosecant are positive

Model answer

Given: Find sin 150°

150° lies in the second quadrant
Reference angle = 180° - 150° = 30°
In second quadrant, sine is positive
sin 150° = sin 30° = 1/2

Therefore, sin 150° = 1/2

Answer structure

• Line 1-2: Find reference angle [1 mark]
• Line 3-4: Apply quadrant rule and calculate [1 mark]

Scoring breakdown

Common mark deductions

• Wrong sign for the quadrant
• Incorrect reference angle calculation

Key phrases to include

• second quadrant
• reference angle = 30°
• sine is positive
• 1/2

Examiner tip

Use the signs of trigonometric functions to identify quadrants: Q2 has sin > 0, cos < 0

Model answer

Given: cos θ = -3/5, sin θ > 0
To find: Quadrant and tan θ

Since cos θ < 0 and sin θ > 0, θ is in the second quadrant.

Using sin²θ + cos²θ = 1:
sin²θ + (-3/5)² = 1
sin²θ + 9/25 = 1
sin²θ = 16/25
sin θ = ±4/5

Since sin θ > 0, we have sin θ = 4/5

tan θ = sin θ/cos θ = (4/5)/(-3/5) = -4/3

Therefore: θ is in quadrant II, and tan θ = -4/3

Answer structure

• Line 1: Identify quadrant using signs [1 mark]
• Line 2-4: Find sin θ using Pythagorean identity [1 mark]
• Line 5: Calculate tan θ correctly [1 mark]

Scoring breakdown

Common mark deductions

• Wrong quadrant identification
• Sign errors in calculations
• Not using the condition sin θ > 0

Key phrases to include

• second quadrant
• Pythagorean identity
• sin θ = 4/5
• tan θ = -4/3

Examiner tip

Always find all solutions within the given range by considering all quadrants where the function is positive

Model answer

Given equation: 2sin θ - 1 = 0
Range: 0° ≤ θ ≤ 360°

2sin θ = 1
sin θ = 1/2

The reference angle where sin θ = 1/2 is 30°
Sine is positive in quadrants I and II

In quadrant I: θ = 30°
In quadrant II: θ = 180° - 30° = 150°

Therefore, θ = 30° or θ = 150°

Answer structure

• Line 1-2: Solve for sin θ [1 mark]
• Line 3: Identify reference angle [1 mark]
• Line 4-5: Find all solutions in given range [1 mark]

Scoring breakdown

Common mark deductions

• Missing one solution
• Not considering the given range
• Incorrect reference angle

Key phrases to include

• sin θ = 1/2
• reference angle = 30°
• quadrants I and II
• θ = 30° or 150°

Examiner tip

Always use exact trigonometric values for special angles and show each algebraic step clearly

Model answer

Given: Triangle ABC with A = 60°, B = 45°, b = 8 units
To find: Side 'a'

Step 1: State the Law of Sines
a/sin A = b/sin B = c/sin C

Step 2: Substitute known values
a/sin 60° = 8/sin 45°

Step 3: Find the values of trigonometric functions
sin 60° = √3/2
sin 45° = √2/2

Step 4: Substitute and solve for 'a'
a/(√3/2) = 8/(√2/2)
a × (2/√3) = 8 × (2/√2)
a × (2/√3) = 16/√2

Step 5: Simplify
a = (16/√2) × (√3/2)
a = 16√3/(2√2)
a = 8√3/√2
a = 8√3/√2 × √2/√2
a = 8√6/2
a = 4√6 units

Therefore, side a = 4√6 units ≈ 9.80 units

Answer structure

• Line 1: State the Law of Sines formula [1 mark]
• Line 2: Substitute given values correctly [1 mark]
• Line 3: Write exact values of trig functions [1 mark]
• Line 4-5: Algebraic manipulation and solving [1 mark]
• Line 6: Final answer in exact form [1 mark]

Scoring breakdown

Common mark deductions

• Using decimal approximations instead of exact values
• Algebraic errors
• Not stating the law clearly
• Incomplete working

Key phrases to include

• Law of Sines
• exact values
• sin 60° = √3/2
• sin 45° = √2/2
• 4√6 units

Examiner tip

For angle of elevation problems, always use tangent ratio when you have the horizontal distance and need height

Model answer

Given: Shadow length = 20 m, Angle of elevation = 35°
To find: Height of the tree

Step 1: Draw and analyze the right triangle

• Base (shadow) = 20 m
• Angle of elevation = 35°
• Height = opposite side

Step 2: Choose appropriate trigonometric ratio
tan 35° = opposite/adjacent = height/shadow
tan 35° = height/20

Step 3: Solve for height
height = 20 × tan 35°
height = 20 × 0.7002 (using calculator)
height = 14.004 m

Step 4: Round to nearest meter
height ≈ 14 m

Therefore, the height of the tree is approximately 14 meters.

Answer structure

• Line 1: Set up the right triangle correctly [1 mark]
• Line 2: Choose correct trigonometric ratio [1 mark]
• Line 3: Calculate using tan 35° [1 mark]
• Line 4: Round to nearest meter as asked [1 mark]

Scoring breakdown

Common mark deductions

• Wrong trigonometric ratio
• Not rounding as requested
• Calculator errors
• Poor diagram or setup

Key phrases to include

• right triangle
• angle of elevation
• tan 35°
• opposite/adjacent
• nearest meter

Examiner tip

Always determine the signs of trigonometric functions based on their quadrants before applying sum formulas

Model answer

Given: sin A = 3/5 (A in QII), cos B = 12/13 (B in QI)
To find: sin(A + B)

Step 1: Find cos A using Pythagorean identity
sin²A + cos²A = 1
(3/5)² + cos²A = 1
9/25 + cos²A = 1
cos²A = 16/25
cos A = ±4/5
Since A is in QII, cos A < 0, so cos A = -4/5

Step 2: Find sin B using Pythagorean identity
sin²B + cos²B = 1
sin²B + (12/13)² = 1
sin²B + 144/169 = 1
sin²B = 25/169
sin B = ±5/13
Since B is in QI, sin B > 0, so sin B = 5/13

Step 3: Apply sum formula
sin(A + B) = sin A cos B + cos A sin B
= (3/5)(12/13) + (-4/5)(5/13)
= 36/65 - 20/65
= 16/65

Therefore, sin(A + B) = 16/65

Answer structure

• Line 1-3: Find cos A using identity and quadrant [2 marks]
• Line 4-5: Find sin B using identity and quadrant [1 mark]
• Line 6-7: Apply sum formula correctly [1 mark]
• Line 8: Calculate final answer [1 mark]

Scoring breakdown

Common mark deductions

• Wrong signs for quadrants
• Incorrect application of sum formula
• Arithmetic errors
• Not using exact fractional form

Key phrases to include

• Pythagorean identity
• quadrant signs
• sum formula
• sin(A + B) = sin A cos B + cos A sin B
• 16/65

Examiner tip

For equations with multiple angles like 2θ, remember to find all solutions in the extended range before dividing

Model answer

Given: cos 2θ = 1/2
Range: 0° ≤ θ ≤ 360°

Step 1: Solve for 2θ
cos 2θ = 1/2
The reference angle where cos = 1/2 is 60°
Cosine is positive in quadrants I and IV

Step 2: Find all values of 2θ in [0°, 720°]
In QI: 2θ = 60°
In QIV: 2θ = 360° - 60° = 300°
For one complete cycle: 2θ = 60°, 300°
For second cycle: 2θ = 60° + 360° = 420°, 300° + 360° = 660°
So 2θ = 60°, 300°, 420°, 660°

Step 3: Solve for θ
θ = 30°, 150°, 210°, 330°

Step 4: Verify all solutions are in [0°, 360°]
All values 30°, 150°, 210°, 330° are within the given range.

Therefore, θ = 30°, 150°, 210°, 330°

Answer structure

• Line 1: Find reference angle for cos 2θ = 1/2 [1 mark]
• Line 2: Find all values of 2θ considering range [1 mark]
• Line 3: Divide by 2 to get θ values [1 mark]
• Line 4: Verify solutions are in given range [1 mark]

Scoring breakdown

Common mark deductions

• Missing solutions
• Not considering the extended range for 2θ
• Incorrect division
• Not verifying final answers

Key phrases to include

• reference angle = 60°
• quadrants I and IV
• extended range for 2θ
• θ = 30°, 150°, 210°, 330°

Examiner tip

For transformed trig functions, apply transformations step by step: amplitude first, then vertical shift

Model answer

Given: f(x) = 3sin(2x - π/4) + 1

Domain: Since sine function is defined for all real numbers, and 2x - π/4 can take any real value, the domain is all real numbers.
Domain = (-∞, ∞) or ℝ

Range: The sine function has range [-1, 1]
Multiplying by 3: 3sin(2x - π/4) has range [-3, 3]
Adding 1: f(x) has range [-3+1, 3+1] = [-2, 4]
Range = [-2, 4]

Therefore: Domain = ℝ, Range = [-2, 4]

Answer structure

• Line 1-2: Identify domain as all real numbers [1 mark]
• Line 3-5: Find range using transformations [1 mark]

Scoring breakdown

Common mark deductions

• Not considering the transformations
• Incorrect interval notation
• Missing amplitude or shift effects

Key phrases to include

• all real numbers
• sine function range [-1, 1]
• amplitude 3
• vertical shift +1
• range [-2, 4]

Do

• Write exact values for special angles
• Use proper mathematical notation
• Box or underline the final answer

Don't

• Don't show unnecessary working
• Don't use decimal approximations unless asked
• Don't forget units if applicable

Do

• State the relevant formula first
• Show one substitution step
• Give final answer with proper conclusion

Don't

• Don't skip the formula statement
• Don't make arithmetic errors
• Don't forget to state what you're finding

Do

• Use proper headings (Given, To Find, Solution)
• Show all intermediate steps
• Apply trigonometric identities correctly
• Consider quadrant signs when applicable

Don't

• Don't skip any major calculation step
• Don't ignore quadrant considerations
• Don't make sign errors

Do

• Number all major steps clearly
• Show complete algebraic manipulation
• Use exact values throughout
• Include verification where possible
• Draw diagrams for application problems

Don't

• Don't rush through calculations
• Don't use shortcuts that skip important steps
• Don't forget to check if answer makes sense
• Don't mix exact and approximate values