If 45° = 1√(2) and 90° = 1, what is the exact value of 2 45° + 3 90° in terms of √(2)?

√(2) + 3. 2 (1√(2)) + 3(1) = 2√(2) + 3 = √(2) + 3.

Grade 10 · Core Mathematics · Lesson 8 of 11

Trigonometric Ratios of Special Angles: 45° and 90°

A Grade 10 Core Mathematics CBE lesson from the sub-strand Trigonometry 1. In it, learners derive the exact trigonometric ratios of 45° using a right-angled isosceles triangle and state the ratios of 90°.

Strand Measurements and GeometrySub-strand Trigonometry 1Lesson code 10CM-24-L840 minutes · 3 slides

KICD learning outcome (d)

Determine trigonometric ratios of special angles: 30°, 45°, 60° and 90° using triangles.

By the end of this lesson, learners should be able to

Derive the exact trigonometric ratios of 45° using a right-angled isosceles triangle and state the ratios of 90°.
Simplify trigonometric expressions involving 30°, 45°, 60°, and 90° without mathematical tables or calculators.
Apply special angle trigonometric ratios to solve geometric problems involving isosceles triangles and right circular cones.

Key vocabulary

Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.; In right-angled isosceles triangle $A B C$ , the hypotenuse $A C$ measures $\sqrt{2}$ units.
Undefined: A mathematical expression that has no valid numeric value, such as dividing by zero.; The value of $\tan 90^{\circ}$ is undefined because it requires dividing one by zero.
Surd: An irrational number written using a root sign that cannot be an exact fraction.; The ratio $\sin 45^{\circ} = \frac{1}{\sqrt{2}}$ contains the surd $\sqrt{2}$ in the denominator.
Slant height: The distance from the apex of a cone along its side to the base circumference.; A cone with slant height $4 \sqrt{2} cm$ has vertical height calculated using cosine.

Lesson preview

Slide 1 · Deriving Ratios of 45° and 90°Opening · 12 min

Figure 1: A right-angled isosceles triangle with legs of 1 unit and a table detailing exact trigonometric ratios for 45 and 90 degrees.

Deriving trigonometric ratios for $45^{\circ}$ relies on a right-angled isosceles triangle with leg lengths of $1$ unit. By Pythagoras' theorem, the hypotenuse is $\sqrt{2}$ . Since the acute angles are $45^{\circ}$ , applying trigonometric definitions yields exact surd expressions. For $90^{\circ}$ , consider an acute angle in a right-angled triangle increasing towards $90^{\circ}$ . The opposite side approaches the hypotenuse in length, so $\sin 90^{\circ} = 1$ . The adjacent side shrinks to $0$ , so $\cos 90^{\circ} = 0$ . $\tan 90^{\circ}$ is undefined. Examine Figure 1 for these exact ratios.

In a right-angled isosceles triangle with legs of length 1, the hypotenuse is $\sqrt{2}$ by Pythagoras' theorem.
The exact trigonometric ratios for $45^{\circ}$ are $\sin 45^{\circ} = \frac{1}{\sqrt{2}}$ , $\cos 45^{\circ} = \frac{1}{\sqrt{2}}$ , and $\tan 45^{\circ} = 1$ .
Scientific calculator verification gives the exact ratios for $90^{\circ}$ as $\sin 90^{\circ} = 1$ and $\cos 90^{\circ} = 0$ .
$\tan 90^{\circ}$ is undefined because $\tan 90^{\circ} = \frac{\sin 90^{\circ}}{\cos 90^{\circ}} = \frac{1}{0}$ and division by zero is not defined.

Worked example. Apply Pythagoras:

X Z^{2} = X Y^{2} + Y Z^{2} = 3^{2} + 3^{2} = 9 + 9 = 18

. Take the square root:

X Z = \sqrt{18} = 3 \sqrt{2} cm

. Calculate

\sin 45^{\circ} = \frac{opposite}{hypotenuse} = \frac{Y Z}{X Z} = \frac{3}{3 \sqrt{2}}

. Divide numerator and denominator by 3 to obtain

\frac{1}{\sqrt{2}}

. Answer: $X Z = 3 \sqrt{2} cm$ and $\sin 45^{\circ} = \frac{1}{\sqrt{2}}$ ..

Check question: If $\cos 45^{\circ} = \frac{1}{\sqrt{2}}$ and $\sin 90^{\circ} = 1$ , what is the exact value of $2 \cos 45^{\circ} + 3 \sin 90^{\circ}$ in terms of $\sqrt{2}$ ?

$\sqrt{2} + 3$
$\frac{1}{\sqrt{2}} + 3$
$2 \sqrt{2} + 3$
$5$

Show answer

$\sqrt{2} + 3$ . $2 (\frac{1}{\sqrt{2}}) + 3 (1) = \frac{2}{\sqrt{2}} + 3 = \sqrt{2} + 3$ .

Slide 2 · Simplifying Exact Trigonometric ExpressionsPractice · 14 min

Figure 2: A summary grid displaying exact trigonometric ratios for special angles alongside three helpful algebraic rules for simplifying surds.

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All 3 slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 10 lessons in this sub-strand

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Trigonometric Ratios of Special Angles: 45° and 90°

By the end of this lesson, learners should be able to

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 11 lessons in Trigonometry 1