Visualizing the core trigonometric concepts based on the Unit Triangle (Hypotenuse = 1 unit).
Adjust Angle ($\theta$):
Angle ($\theta$): °
Base (Adjacent) Length:
0.000
Since Hypotenuse = 1, Base Length = $\cos(\theta)$
Perpendicular (Opposite) Length:
0.000
Since Hypotenuse = 1, Perpendicular Length = $\sin(\theta)$
**Louisiana Standard Connection:** This model demonstrates that for any right triangle with a fixed hypotenuse (like a Unit Circle radius), the side lengths are determined only by the angle ($\theta$). Specifically, $Base = \cos(\theta)$ and $Perpendicular = \sin(\theta)$.
Calculate the perimeter of polygons by plotting vertices in the coordinate plane (LA Math Standard 6.G.A.3).
Enter coordinates (one point per line). Max 8 points.
Total Perimeter: 0 units
Please ensure all points have the same X or Y coordinate to use the absolute difference rule (6.G.A.3).
Use geometric rules to find unknown angles based on fundamental relationships.
**Louisiana Standard Connection:** This section focuses on using facts about supplementary and angles in a triangle to write and solve equations for unknown angles in a figure.
The canvas updates based on the calculator you are currently adjusting.
Input two known angles (A and B) to find the missing third angle (C).
Calculated Angle C:
50°
Angles A and B must sum to less than 179°.
Input the known angle ($\alpha$) to find the missing supplementary angle ($\beta$).
Calculated Missing Angle ($\beta$):
55°
**Rule:** Angles that form a straight line add up to $180^\circ$.
Use algebraic expressions for angle relationships (Vertical and Supplementary) to solve for $x$ and find the angle measure. (LA Math Standard 7.G.B.5)
**Louisiana Standard Connection (7.G.B.5):** This section demonstrates how to use the facts about angle pairs to set up and solve equations (e.g., $A=B$ or $A+B=180$).
Visualization reflects the results of the **Vertical Angles** calculator below.
Set expressions equal: $(A x + B) = (C x + D)$
Calculated $x$ Value:
$x = $ 30.00
Angle Measure:
75.00°
Set sum equal to $180^\circ$: $(A x + B) + (C x + D) = 180$
Calculated $x$ Value:
$x = $ 27.00
Angle 1:
118.00°
Angle 2:
62.00°
Explore similar right triangles to see that the ratio of corresponding sides is **constant** for a given angle, regardless of the triangle's size.
**Louisiana Standard Connection (Pre-SOH CAH TOA):** This demonstrates the geometric definition of trigonometric ratios based on similarity—the foundation for defining sine, cosine, and tangent.
Adjust Scale Factor:
Scale Factor (k): 1.5
Adjacent Side ($b_1$): 100.00
Opposite Side ($a_1$): 100.00
Hypotenuse ($c_1$): 141.42
Ratio A/B ($\tan(\theta)$): 1.000
Adjacent Side ($b_2$): 150.00
Opposite Side ($a_2$): 150.00
Hypotenuse ($c_2$): 212.13
Ratio A/B ($\tan(\theta)$): 1.000
Conclusion:
Since Angle $\theta$ is fixed, the ratio $Opposite/Adjacent$ is **constant** ($\frac{a_1}{b_1} = \frac{a_2}{b_2}$), proving side proportionality in similar triangles.
Use scale factor to determine the dimensions of similar figures (LA Math Standard, typically 7.G.A.1).
**Louisiana Standard Connection:** This demonstrates using the concept of scale factor ($k$) and proportions ($\frac{\text{New Length}}{\text{Original Length}} = k$) to find an unknown length in a scaled version of a figure.
The relationship is: $\frac{W_B}{W_A} = \frac{H_B}{H_A} = k$
Calculated Scale Factor (k):
1.50
Unknown Width ($W_B$):
60.00
Explore the fundamental relationship between the sides of a right triangle: $\text{a}^2 + \text{b}^2 = \text{c}^2$.
**Louisiana Standard Connection (8.G.B.7):** This demonstrates applying the Pythagorean theorem to determine unknown side lengths in right triangles.
Input two known sides. Leave the unknown side blank or as '?' to solve.
$3^2 + 4^2 = \text{c}^2$
Missing Side Length:
5.00
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Use the Pythagorean Theorem to find the distance between two points in a coordinate plane: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
**Louisiana Standard Connection (8.G.B.8):** This demonstrates finding the distance between two points in a coordinate system using the distance formula, which is an application of the Pythagorean Theorem. Drag the points to see the values change!
x1: 0.00
y1: 0.00
x2: 0.00
y2: 0.00
Change in x ($\Delta x = x_2 - x_1$): 0.00
Change in y ($\Delta y = y_2 - y_1$): 0.00
Distance Squared ($d^2 = \Delta x^2 + \Delta y^2$): 0.00
Final Distance ($d$):
0.00
Determine if a triangle with given side lengths forms a right, acute, or obtuse triangle by comparing $a^2 + b^2$ to $c^2$.
**Louisiana Standard Connection (Converse):** This is the application of the Pythagorean Theorem to test if a given triangle is a right triangle. If $a^2 + b^2 = c^2$, it is right. If $a^2 + b^2 > c^2$, it is acute. If $a^2 + b^2 < c^2$, it is obtuse.
The longest side will be tested as $c$.
Sum of Squares ($a^2 + b^2$): 25.00
Longest Side Squared ($c^2$): 25.00
Conclusion:
Right Triangle
Demonstrate that trigonometric ratios depend ONLY on the angle ($\theta$), not the size of the triangle. (G-SRT.C.6)
**Louisiana Standard Connection (G-SRT.C.6):** This visually proves that $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$ are defined by ratios of sides in right triangles and are constant for a given angle due to similarity.
Adjust Angle ($\theta$):
Angle ($\theta$): 40°
Adjust Hypotenuse Length (Scale):
Hypotenuse (H): 100 units
Opposite (O): 0.00
Adjacent (A): 0.00
Hypotenuse (H): 0.00
Sine ($\sin(\theta) = \frac{O}{H}$):
0.0000
Cosine ($\cos(\theta) = \frac{A}{H}$):
0.0000
Tangent ($\tan(\theta) = \frac{O}{A}$):
0.0000