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)

Consider the following triangle.

a right triangle with a nonright angle of 67 degrees, a hypotenuse of x units and an opposite leg of y units

Determine the approximate value of

sin 6 7^{\circ}

without the use of a calculator. We have been given the following information.

sin 2 3^{\circ} cos 2 3^{\circ} tan 2 3^{\circ} \approx 0.391 \approx 0.921 \approx 0.424

In the given right triangle, we know two angles. This means we can find the triangle's third angle by using the Interior Angles Theorem. m∠ θ+67^(∘) +90^(∘)= 180^(∘) ⇓ m∠ θ= 23^(∘) As we can see, the third angle has a measure of 23^(∘). Let's add this information to the diagram.

We want to know the value of sin 67^(∘). This is the ratio of the opposite leg to the hypotenuse. Using the information in the diagram, we can write an equation containing sin 67^(∘).

We can also write a second equation containing cos 23^(∘) by using the cosine ratio.

Since both sin 67^(∘) and cos 23^(∘) equals yx, we can equate these expressions and then use the information in the exercise about cos 23^(∘).

Consider $△ A B C .$

a right triangle where the sine value and cosine value of C and A are given as expressions

What is the value of

x ?

What is the length of

\overline{A B}

B C \approx 6.928 ?

Round to a whole number of units.

Examining the right triangle, we notice that ∠ A and ∠ C represent the acute angles of the triangle. Therefore, by the Interior Angles Theorem, the sum of their measures equals 90^(∘). m∠ A+m∠ C+90^(∘)=180^(∘) ⇓ m∠ A+m∠ C=90^(∘) Since the cosine of an acute angle equals the sine of the complement of the angle, we know that cos A must equal sin C. Therefore, we get the following equation. cos A&=sin C &⇓ 2x-11/2&=7/2-x Let's solve this equation for x.

From Part A, we know the value of x. Next, let's substitute its value into the expression for sin C.

By using the inverse of the sine ratio we can find the measure of ∠ C. sin C=1/2 ⇕ m∠ C=sin^(- 1)1/2=30^(∘) Let's add m∠ C and the length of BC to the diagram.

By using the tangent ratio we can now find AB.

As we can see, the length of AB is about 4 units.

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