Unlocking Sine and Cosine of Complementary Angles

	$sin θ$	$cos θ$
$θ = 1 5^{\circ}$	$0.258819 \dots$	$0.965925 \dots$
$θ = 3 0^{\circ}$	$0.5$	$0.866025 \dots$
$θ = 4 5^{\circ}$	$0.707106 \dots$	$0.707106 \dots$
$θ = 6 0^{\circ}$	$0.866025 \dots$	$0.5$
$θ = 7 5^{\circ}$	$0.965925 \dots$	$0.258819 \dots$

sin θ

cos θ

θ = 1 5^{\circ}

0.258819 \dots

0.965925 \dots

θ = 3 0^{\circ}

0.5

0.866025 \dots

θ = 4 5^{\circ}

0.707106 \dots

0.707106 \dots

θ = 6 0^{\circ}

0.866025 \dots

0.5

θ = 7 5^{\circ}

0.965925 \dots

0.258819 \dots

Cofunction Identities
$sin θ = cos (9 0^{\circ} - θ)$	$cos θ = sin (9 0^{\circ} - θ)$
$tan θ = cot (9 0^{\circ} - θ)$	$cot θ = tan (9 0^{\circ} - θ)$
$sec θ = csc (9 0^{\circ} - θ)$	$csc θ = sec (9 0^{\circ} - θ)$

Cofunction Identities

sin θ = cos (9 0^{\circ} - θ)

cos θ = sin (9 0^{\circ} - θ)

tan θ = cot (9 0^{\circ} - θ)

cot θ = tan (9 0^{\circ} - θ)

sec θ = csc (9 0^{\circ} - θ)

csc θ = sec (9 0^{\circ} - θ)

Below we see $△ A B C .$

Triangle ABC with two legs of 5 and 9 units and a hypotenuse of 10.3 units.

Which statement(s) are true when

sin B

approaches

1 ?

i : i i i : v : A C \approx B C A C < B C A B > A C i i : m ∠ B > m ∠ C i v : m ∠ C > m ∠ B v i : A C > A B

In a right triangle, sine is the ratio of an acute angle's opposite side to the hypotenuse. sin θ = Opposite/Hypotenuse As a consequence of this ratio approaching 1, the opposite side of ∠ B would then increase. That then makes the difference between the hypotenuse and the opposite side smaller. Let's illustrate what happens to △ ABC as the opposite side of B increases.

Notice how the ratio approaches 1 as AC increases. This means that AC comes increasingly closer to BC. Therefore, we know that the first statement is true. i: AC≈ BC ✓ However, the quotient will never equal 1 since the hypotenuse is always greater than either of the legs in a right triangle. Therefore, the third statement must also be true. iii: AC < BC ✓ We also notice that ∠ B increases and ∠ C decreases as AC increases. This means m∠ B>m∠ C. When we examine the statements, we see that the second is true and the fourth is false. ii:& m∠ B > m∠ C ✓ iv:& m∠ C < m∠ B * For sin B to approach 1, the side opposite B needs to have a length close to that of the hypotenuse. That can only be the case if the opposite side is greater that the adjacent side. Now we know that the fifth statement is false and the sixth is true. v:& AB > AC * vi:& AC > AB ✓ We can now summarize that statements i, ii, iii, and vi are correct.

Properties of the Trigonometric Ratios

Catch-Up and Review

Investigating the Relationship Between Sine and Cosine

Sine and Cosine of Complementary Angles

Proof

Convert Between Sine and Cosine

Hint

Solution

Practice Converting Between Sine and Cosine

Investigating Properties of Right Triangles

Answer

Hint

Solution

Solving Problems Using the Relationship Between Sine and Cosine

Hint

Solution

Using the Relationship Between Sine and Cosine to Solve Problems

Hint

Solution

Tangent and Cotangent of Complementary Angles

Proof

Solving Problems Using the Relationship Between Tangent and Cotangent

Hint

Solution

Calculating Distances Using Trigonometry

Hint

Solution

Secant and Cosecant Ratios

Properties of the Trigonometric Ratios

Recommended exercises

	12 Theory slides
	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions