Q: Write the following function in terms of its cofunction. Assume that all angles in which an unknown appears are acute angles. sin (θ + 25°) (Simplify your answer. Do not include the degree symbol in your answer.)

The correct answer is: $\cos(65-\theta)$ Explanation We use the cofunction identity for sine and cosine (angles in degrees):$\sin x=\cos(90-x)$. Steps: Let $x=\theta+25$. Apply the identity: $$\sin(\theta+25)=\cos(90-(\theta+25))$$ Simplify inside the cosine: $$\cos(90-(\theta+25))=\cos(90-\theta-25)=\cos(65-\theta)$$ Therefore $\sin(\theta+25)=\cos(65-\theta)$. (As a check, using the evenness of cosine one may also write $\cos(65-\theta)=\cos(\theta-65)$, so both forms are equivalent.) Summary: By the cofunction identity $\sin x=\cos(90-x)$ and direct substitution $x=\theta+25$, the expression simplifies to $\cos(65-\theta)$ (degrees implied; the final answer omits the degree symbol as requested).

Q: Are the two triangles congruent?

I need more information or a diagram to determine if the two triangles are congruent. Explanation Triangles are congruent if they have exactly the same size and shape, which can be proven using criteria such as SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), or HL (Hypotenuse-Leg for right triangles). Without specific measurements or a diagram, I cannot definitively answer whether the triangles are congruent. Please provide: A diagram of the two triangles, or The lengths of sides and measures of angles, or Any given congruence criteria or markings indicating equal sides/angles. Once I have that information, I can determine if the triangles are congruent.

Q: Which equation has the least steep graph?

The equation with the least steep graph is the one with the smallest absolute value of slope. Answer: The equation with the smallest absolute value of the slope coefficient. Explanation: The steepness of a graph, especially for linear equations, is determined by its slope ($m$). The smaller the absolute value of $m$, the less steep (flatter) the line appears. Steps: Identify the slope ($m$) in each equation. Compare the absolute values of these slopes. The equation with the smallest $|m|$ has the least steep graph. For example, if the options are: $y = 2x + 3$ (slope $m=2$) $y = -0.5x + 1$ (slope $m=-0.5$) $y = x - 4$ (slope $m=1$) $y = 0.1x + 5$ (slope $m=0.1$) The least steep graph corresponds to the smallest $|m|$, which is $0.1$ in this case. Therefore, the equation $y = 0.1x + 5$ has the least steep graph. If you provide specific equations, I can compare them directly!

Q: If ∠PORT measures 75°, what is the measure of ∠SQR?

The measure of ∠SQR is 75°. Explanation Since the problem states that ∠PORT measures 75°, and asks for the measure of ∠SQR, it suggests that these angles are related in some way—likely they are either equal or supplementary based on the geometric configuration. Without a diagram, the most common scenario is that ∠PORT and ∠SQR are either vertically opposite angles, corresponding angles, or part of the same geometric figure where angles are equal. Given typical geometric conventions, if ∠PORT and ∠SQR are corresponding or vertically opposite angles in a figure, then: Steps: Recognize the relationship: If the angles are corresponding or vertically opposite, then they are equal. Therefore, $\boxed{\text{∠SQR} = 75°}$. Therefore, the measure of ∠SQR is 75°.

Q: How do you write the number 314,207 in expanded form?

The number 314,207 in expanded form is: $300,000 + 10,000 + 4,000 + 200 + 0 + 7$. Explanation To write a number in expanded form, you break it down into the sum of each digit multiplied by its place value. The place values from left to right are hundreds of thousands, ten thousands, thousands, hundreds, tens, and ones. Steps: The digit 3 is in the hundreds of thousands place: $3 \times 100,000 = 300,000$ The digit 1 is in the ten thousands place: $1 \times 10,000 = 10,000$ The digit 4 is in the thousands place: $4 \times 1,000 = 4,000$ The digit 2 is in the hundreds place: $2 \times 100 = 200$ The digit 0 is in the tens place: $0 \times 10 = 0$ The digit 7 is in the ones place: $7 \times 1 = 7$ Adding these together gives the expanded form: $$300,000 + 10,000 + 4,000 + 200 + 0 + 7$$ Therefore, the expanded form of 314,207 is $300,000 + 10,000 + 4,000 + 200 + 0 + 7$.

Q: Find the volume of the triangular prism.

$$V=60 cm^{3}$$ Explanation Calculate the area of the triangular base by using the formula for the area of a triangle (A = 0.5 * base * height). Area = 0.5 * 4 cm * 6 cm = 12 cm². Calculate the volume of the prism by multiplying the area of the base by the height (V = Area * height). Volume = 12 cm² * 5 cm = 60 cm³.

Question

Write the following function in terms of its cofunction. Assume that all angles in which an unknown appears are acute angles. sin (θ + 25°) (Simplify your answer. Do not include the degree symbol in your answer.)

Accepted Answer

The correct answer is: $\cos(65-	heta)$

Explanation

We use the cofunction identity for sine and cosine (angles in degrees):$\sin x=\cos(90-x)$.

Steps:

Let $x=	heta+25$. Apply the identity:

$$\sin(	heta+25)=\cos(90-(	heta+25))$$

Simplify inside the cosine:

$$\cos(90-(	heta+25))=\cos(90-	heta-25)=\cos(65-	heta)$$

Therefore $\sin(	heta+25)=\cos(65-	heta)$. (As a check, using the evenness of cosine one may also write $\cos(65-	heta)=\cos(	heta-65)$, so both forms are equivalent.)

Summary: By the cofunction identity $\sin x=\cos(90-x)$ and direct substitution $x=	heta+25$, the expression simplifies to $\cos(65-	heta)$ (degrees implied; the final answer omits the degree symbol as requested).