Grade 10 Mathematics: Trigonometry
Comprehensive Study Guide
This study guide covers everything you need to master Grade 10 trigonometry — from the foundations of right-angled
triangles and Pythagoras' theorem, through to trigonometric ratios, special angles, the Cartesian plane, graphs, and real-
world applications. Work through each section carefully, use the practice exercises to test your understanding, and refer
back to the theory whenever you need a refresher.
01 02
Section 1–2 Section 3–4
Pythagoras & trig ratios Special angles & the Cartesian plane
03 04
Section 5–6 Section 7+
Elevation, depression & graphs Practice exercises & solutions
© E-Loné Scheepers 2026
, SECTION 1
Pre-Trig Foundations & Right-Angled Triangles
Before we dive into trigonometry itself, it is essential to be completely comfortable with the language and structure of
right-angled triangles. Every trigonometric calculation you will ever do in Grade 10 is built on identifying the sides of a
right-angled triangle correctly relative to a chosen reference angle. Get this foundation right, and everything else follows
naturally.
A right-angled triangle has three sides, each with a specific name. The names of two of those sides depend on which angle
you are focusing on — so always start a problem by marking your reference angle clearly before you label any sides.
Hypotenuse Opposite Adjacent
Always opposite the 90° angle. The side directly across from your The side directly next to your
Always the longest side. Its reference angle θ. Draw an arrow reference angle (not the
position never changes regardless from the angle — the side it points hypotenuse). It forms one of the
of which angle you choose. to is Opposite. two "arms" of the angle alongside
the hypotenuse.
If you switch your reference angle from one acute angle to the other in the same triangle, the Opposite and
Adjacent sides swap — but the Hypotenuse stays the same.
© E-Loné Scheepers 2026
, SECTION 1
The Theorem of Pythagoras
Pythagoras' theorem is the single most important result in all of plane geometry. It tells us that in any right-angled
triangle, the square of the hypotenuse is always equal to the sum of the squares of the other two sides. This gives us the
power to find any missing side as long as we know the other two.
The Core Formula Pythagorean Triples
Main form: These are sets of whole numbers that satisfy the theorem
exactly — no rounding needed! Memorise these and
2 2 2
c =a +b always check if a triangle fits one of these patterns before
you start calculating.
Rearranged to find a:
3 : 4 : 5 (and multiples: 6:8:10, 9:12:15, …)
2 2 2
a =c −b 5 : 12 : 13
Rearranged to find b: 8 : 15 : 17
7 : 24 : 25
2 2 2
b =c −a
© E-Loné Scheepers 2026
, SECTION 2
The Three Primary Trigonometric Ratios
Trigonometry is fundamentally about linking the angles of a right-angled triangle to the ratios of its side lengths. For any
acute angle θ, there are three primary ratios. Every trig problem you encounter will use at least one of these, so you must
know them perfectly.
Sine Cosine Tangent
Opposite y Adjacent x Opposite y
sin θ = = cos θ = = tan θ = =
Hypotenuse Hypotenuse Adjacent
r x
r
🧠 Mnemonic — Oh Hell Another Hour Of Algebra:
Opposite / Hypotenuse → sin | Adjacent / Hypotenuse → cos | Opposite / Adjacent → tan
© E-Loné Scheepers 2026
Comprehensive Study Guide
This study guide covers everything you need to master Grade 10 trigonometry — from the foundations of right-angled
triangles and Pythagoras' theorem, through to trigonometric ratios, special angles, the Cartesian plane, graphs, and real-
world applications. Work through each section carefully, use the practice exercises to test your understanding, and refer
back to the theory whenever you need a refresher.
01 02
Section 1–2 Section 3–4
Pythagoras & trig ratios Special angles & the Cartesian plane
03 04
Section 5–6 Section 7+
Elevation, depression & graphs Practice exercises & solutions
© E-Loné Scheepers 2026
, SECTION 1
Pre-Trig Foundations & Right-Angled Triangles
Before we dive into trigonometry itself, it is essential to be completely comfortable with the language and structure of
right-angled triangles. Every trigonometric calculation you will ever do in Grade 10 is built on identifying the sides of a
right-angled triangle correctly relative to a chosen reference angle. Get this foundation right, and everything else follows
naturally.
A right-angled triangle has three sides, each with a specific name. The names of two of those sides depend on which angle
you are focusing on — so always start a problem by marking your reference angle clearly before you label any sides.
Hypotenuse Opposite Adjacent
Always opposite the 90° angle. The side directly across from your The side directly next to your
Always the longest side. Its reference angle θ. Draw an arrow reference angle (not the
position never changes regardless from the angle — the side it points hypotenuse). It forms one of the
of which angle you choose. to is Opposite. two "arms" of the angle alongside
the hypotenuse.
If you switch your reference angle from one acute angle to the other in the same triangle, the Opposite and
Adjacent sides swap — but the Hypotenuse stays the same.
© E-Loné Scheepers 2026
, SECTION 1
The Theorem of Pythagoras
Pythagoras' theorem is the single most important result in all of plane geometry. It tells us that in any right-angled
triangle, the square of the hypotenuse is always equal to the sum of the squares of the other two sides. This gives us the
power to find any missing side as long as we know the other two.
The Core Formula Pythagorean Triples
Main form: These are sets of whole numbers that satisfy the theorem
exactly — no rounding needed! Memorise these and
2 2 2
c =a +b always check if a triangle fits one of these patterns before
you start calculating.
Rearranged to find a:
3 : 4 : 5 (and multiples: 6:8:10, 9:12:15, …)
2 2 2
a =c −b 5 : 12 : 13
Rearranged to find b: 8 : 15 : 17
7 : 24 : 25
2 2 2
b =c −a
© E-Loné Scheepers 2026
, SECTION 2
The Three Primary Trigonometric Ratios
Trigonometry is fundamentally about linking the angles of a right-angled triangle to the ratios of its side lengths. For any
acute angle θ, there are three primary ratios. Every trig problem you encounter will use at least one of these, so you must
know them perfectly.
Sine Cosine Tangent
Opposite y Adjacent x Opposite y
sin θ = = cos θ = = tan θ = =
Hypotenuse Hypotenuse Adjacent
r x
r
🧠 Mnemonic — Oh Hell Another Hour Of Algebra:
Opposite / Hypotenuse → sin | Adjacent / Hypotenuse → cos | Opposite / Adjacent → tan
© E-Loné Scheepers 2026
