Shop Essentials (Applied Mathematics)

Trigonometry: Sine Bar Applications 221

Trigonometry: Sine Bar Applications discusses sine bars and the trigonometry required to use them. Sine bars are used when an angle needs to be machined, measured, or inspected. Sine bars are used with gage blocks to set a workpiece at an angle. To find the necessary measurements for the gage blocks or the sine bar angle, trigonometric ratios are used. These ratios include sine, cosine, and tangent. Gage pins are sometimes used with sine bars and gage blocks to increase the range of measurements.

After taking this class, a user should be able to make the necessary calculations for setting up a specific workpiece angle using a sine bar.

  • Difficulty Intermediate

  • Format Online

  • Number of Lessons 16

  • Language English


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Course Outline
  • The Sine Bar
  • Sine Bar Characteristics
  • The Purpose of the Sine Bar
  • Setting the Sine Bar
  • Sine Bar Review
  • The Sine Bar and Trig Ratios
  • Determining Gage Block Size
  • Determining Gage Block Size: In Action
  • Determining the Sine Bar Angle
  • Determining the Sine Bar Angle: In Action
  • The Sine Bar and Trig Ratios Review
  • Gage Pins
  • Gage Pin Distances
  • Checking Dimensions with a Gage Pin
  • Checking Dimensions with a Gage Pin: In Action
  • Final Review
  • Define sine bar.
  • Describe sine bar characteristics.
  • Distinguish between sine bar applications in inspection and in machining.
  • Explain how to set up a sine bar.
  • Define sine, cosine, and tangent. Explain how trigonometry relates to the sine bar.
  • Describe how to find the gage block size for a specific sine bar angle.
  • Calculate the gage block size for a specific sine bar angle.
  • Describe how to find the measurement of the sine bar angle using a gage block height.
  • Calculate the measurement of the sine bar angle using a gage block height.
  • Describe gage pins.
  • Describe how to use a gage pin to find missing distances on a part.
  • Explain how to find missing dimensions with a sine bar and a gage pin.
  • Calculate missing dimensions using a sine bar and gage pin.
Vocabulary Term


Conforming exactly to a given standard. An accurate sine bar gives the correct measurements.


Having a common vertex or a common side. In a right triangle, the adjacent side is the side next to the reference angle.


The combination of either two rays with the same endpoint or two straight lines that intersect. Angles are measured in degrees, which specify the amount of separation between the sides of the angle.


An imaginary straight line passing through the center of an object. Two or more axes are perpendicular to one another.

center distance

The horizontal length from the center of one cylindrical roller to the center of the other roller on a sine bar. A sine bar's size refers to its center distance.


A workholding operation that secures a workpiece against locators. Clamping resists the other forces that occur during machining.

complementary angles

Two angles that together measure exactly 90 degrees. For example, angles measuring 52 degrees and 38 degrees are complementary angles.

corresponding angles

Two angles that are located in the same relative location. If a transversal intersects two parallel lines, corresponding angles appear on the same side of the transversal.


In a right triangle, the ratio of the length of the side adjacent to an angle divided by the hypotenuse. The cosine of an angle is a trigonometric ratio.


The distance from one edge of a circle to the opposite edge through the center. A diameter divides a circle in half.


A mathematical statement that shows how values relate to each other. Both sides of an equation show different ways to express the same value.

gage blocks

A set of rectangular metal devices used to measure very precise dimensions. Gage blocks can be combined to create a wide variety of sizes.

gage pin

A cylindrical pin often used with the sine bar to help determine a range of precise measurements. A gage pin provides an easily accessible surface and a known height from which to measure all distances and edges.

gage pin height

The distance from the surface plate to the top edge of the gage pin. Gage pin height acts as a starting point for calculating and measuring distances.


A dense, hard type of rock that exhibits excellent wear resistance and stability. Granite is often used for inspection surfaces.


The use of an abrasive to wear away at the surface of a workpiece and change its shape. Grinding achieves highly accurate measurements.

height gage

A type of measuring instrument with a precision finished base, a beam that is at a right angle to the base, and an indicator. Height gages are used to measure gage pin height.


The longest side of a right triangle. The hypotenuse is always opposite the 90° angle in a right triangle.


The examination of a part during or after its creation to confirm that it adheres to specifications. Inspection involves measuring parts with devices such as the sine bar.

inverse sine

The function on a calculator that gives the angle in degrees for a known sine value. Entering a value into a calculator and pressing the inverse sine button will show what angle has a sine of that value.


A precision abrasion process used to bring a surface to a desired state of finish or tolerance. Lapping uses a fine abrasive to grind away small amounts of material from a workpiece.


Remove material to form or finish a part. Sine bars are used to machine angled parts.


Across from a given point or angle. In a right triangle, the opposite side does not share a vertex with the reference angle.


Extending in the same direction and equally distant from one another at all points. Parallel lines never intersect.


Able to give the same exact measurements for a dimension repeatedly over time. A precise sine bar gives consistent measurements.


The distance from the center to any point on the edge of a circle. The radius is half the diameter.

reference angle

A measured or known angle in a right triangle. The reference angle cannot be the triangle's 90 degree angle.

right triangle

A triangle with one angle that measures exactly 90 degrees. The other two angles in a right triangle must total exactly 90 degrees.


The series of tasks necessary to prepare a part for machining. Setup includes positioning the part and holding it in place.


In a right triangle, the ratio of the length of the side opposite an angle divided by the hypotenuse. The sine of an angle is a trigonometric ratio.

sine bar

A steel bar with a matching cylinder on each end that is used for setting up work at various angles for the machining or inspection of parts. Sine bars are commonly made of tool steel.

surface plate

A hard, flat surface, usually made of granite, which is used as a base for inspecting parts. Surface plates provide the flat surface for sine bars to rest on.


In a right triangle, the ratio of the length of the side opposite an angle divided by the adjacent side. The tangent of an angle is a trigonometric ratio.


The unwanted but acceptable deviation from the desired dimension. Tighter tolerances allow for less deviation.

tool steel

A specialized type of alloy steel that exhibits excellent strength, toughness, and wear resistance. Sine bars are commonly made of tool steel.


A line that intersects two or more lines at different points. A transversal forms several different angles with each line that it intersects.

trigonometric ratios

Ratios that describe the relationships between the sides and angles of a right triangle. Trigonometric ratios include sine, cosine, and tangent.


The branch of mathematics that addresses the measurements and relationships of a triangle and its parts. Trigonometry is often used when working with a sine bar.


A point where two lines or line segments meet or intersect. A triangle has three vertices.


Durable. A wear-resistant surface or material is not easily damaged by normal wear or usage.

work hardening

Bending or shaping a metal to increase its hardness. Work hardening also increases the strength of the metal.


The process of securely supporting, locating, and clamping a workpiece for a manufacturing operation. Workholding devices include vises, jigs, and clamps.


A part or component being worked on during manufacturing. Workpieces are placed on sine bars to machine or measure angles.