Unlock Trigonometric Identities with Double-Angle and Half-Angle Formulas

Find the exact value of the following expressions given that $cos θ = \frac{1 5}{1 7}$ and $0^{\circ} < θ < 9 0^{\circ} .$

sin 2 θ

tan 2 θ

We are asked to find the value of sin 2θ. To do so, let's start by recalling the Double-Angle Identity for sine. sin 2θ=2sinθcosθ We are given the value of cosθ. However, we do not know the value of sin θ. In order to find it, we can use one of the Pythagorean Identities. sin^2 θ+cos^2 θ=1 Let's substitute cosθ with 1517 and solve the equation for sinθ.

We found that sin θ =± 817. To find the sign of sin θ, recall that θ is greater than 0^(∘) and less than 90^(∘). This means that if we draw the angle θ in standard position, the terminal side will be located in Quadrant I.

We can see that sin θ is positive in Quadrant I. sin θ = 8/17 Now that we know both sinθ and cosθ, we can substitute these two values into the Double-Angle Identity and calculate sin 2θ.

We are asked to find the value of tan 2θ. First, let's recall the Double-Angle Identity for tangent. tan 2θ=2tan θ/1-tan^2 θ As we can see, we need to find the value of tanθ. To do so, we can use the Tangent Identity. tanθ=sinθ/cosθ Let's substitute 817 for sinθ and 1517 for cosθ into the equation.

Now that the value of tanθ is known, we can calculate tan2θ.

Find the exact value of the following expressions given that $tan θ = \frac{4}{3}$ and $π < θ < \frac{3 π}{2} .$

sin \frac{θ}{2}

cos \frac{θ}{2}

We are asked to find the value of sin θ2. To do so, let's recall the Half-Angle Identity for sine. sin θ2=± sqrt(1-cosθ/2) First, we need to find the value of cosθ. We know that tanθ= 43, so let's use this piece of information. Recall one of the Pythagorean Identities that involves tangent. 1+tan^2 θ=sec^2 θ There is also a Reciprocal Identity that describes the relationship between secant and cosine. sec θ=1/cosθ By substituting 1cosθ for secθ into the mentioned Pythagorean Identity, an equation that relates tangent and cosine can be obtained. 1+tan^2 θ=( 1/cosθ)^2 Now, we can substitute the known value of tanθ and solve the equation for cosθ.

We found that cos θ =± 35. Now, we need to determine the sign of cos θ. Recall that θ is greater than π and less than 3π2. This means that if we draw the angle θ in standard position, the terminal side will be located in Quadrant III.

We can see that cos θ is negative for θ in Quadrant III. cos θ =- 3/5 We will substitute the value of cosθ into the Half-Angle Identity and calculate sin θ2.

Finally, we need to determine the sign of sin θ2. Knowing that θ is greater than π and less than 3π2, we can find the range of values for θ2. π< θ < 3π/2 ⇔ π2 < θ/2 < 3π/4 Since θ2 is greater than π2 and less than 3π4, it is located in Quadrant II.

Therefore, sin θ2 is positive. sin θ/2= 2sqrt(5)/5

Here, we are asked to find the value of cos θ2. Let's start by recalling the Half-Angle Identity for cosine. cosθ/2=± sqrt(1+cosθ/2) In Part A, we found that cosθ=- 35. Let's substitute this value into the identity and calculate cos θ2.

Previously, we found that θ2 is located in Quadrant II, where the cosine is negative. Therefore, cos θ2 is negative. cos θ/2= - sqrt(5)/5

Use a Double-Angle Identity to find the exact value of each expression.

sin 30 0^{\circ}

cos 45 0^{\circ}

We are asked to use a Double-Angle Identity to find the exact value of sin 300^(∘). Let's recall the Double-Angle Identity that involves sine. sin 2θ = 2 sin θ cos θ Now, we can use this formula to find the value of sin 300^(∘). We will start by rewriting 300^(∘) as a product where one of the factors is 2.

According to the table of trigonometric values of common angles, sin 150^(∘) = 12 and cos 150^(∘) = - sqrt(3)2. Therefore, we can substitute these two values into our expression.

We want to use a Double-Angle Identity to find the exact value of cos 450^(∘). Let's recall the Double-Angle Identity that involves cosine. cos 2θ = 2 cos^2 θ - 1 To use this formula, we need to write 450^(∘) as a product where one factor is 2.

By using the table of trigonometric values of notable angles, we can recall that cos 225^(∘) = - sqrt(2)2. Let's substitute this value into our expression and evaluate it.

Use a Half-Angle Identity to find the exact value of each expression.

cos 1 5^{\circ}

tan 22 . 5^{\circ}

We are asked to use a Half-Angle Identity to find the exact value of cos 15^(∘). First, let's review the Half-Angle Identity that involves cosine. cos θ/2 = ± sqrt(1+cos θ/2) Next, we can use this formula to find the value of cos 15^(∘). We will start by rewriting 15^(∘) as a quotient of some number and 2.

According to the table of trigonometric values of common angles, cos 30^(∘) = sqrt(3)2. Therefore, we can substitute this value into our expression.

Finally, we need to determine the sign of the expression. To do so, let's recall the signs of sine, cosine, and tangent in the four quadrants of the coordinate plane.

Since 15^(∘) is located in Quadrant I, we know that cos 15^(∘) is positive. cos 15^(∘)= sqrt(2+sqrt(3))/2

We need to find the exact value of tan 22.5^(∘). To do so, let's recall the Half-Angle Identity for tangent. tan θ/2=± sqrt(1-cosθ/1+cosθ) First, we will express 22.5 as the quotient of some number and 2.

From the table of trigonometric values of notable angles, cos 45^(∘) = sqrt(2)2. Therefore, we can substitute this value into our expression.

Finally, we will determine the sign. To do so, let's recall the signs of sine, cosine, and tangent in the four quadrants of the coordinate plane.

Since 22.5^(∘) is located in Quadrant I, we can conclude that tan 22.5^(∘) is positive. tan 22.5^(∘)= sqrt(3 - 2sqrt(2))

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

Double and Half-Angle Identities

Catch-Up and Review

Proof

Sine Identity

Cosine Identities

Tangent Identity

Extra

Hint

Solution

Hint

Solution

Proof

Sine Identity

Cosine Identity

Tangent Identity

Extra

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Using the Half-Angle Identity

Using the Double-Angle Identity

Hint

Solution

Hint

Solution

Double and Half-Angle Identities

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